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b(Lie)e(group)g(of)h(scale)g(transformations)800 12715 y FB(x)1433 12313 y Fq(0)2051 12715 y FA(=)307 b FB(\025)184 b(x)p FA(,)370 b FB(\025)307 b Fz(2)g Fy(R)8312 12881 y Fu(+)9047 12715 y FA(,)369 b(is)g FB(E)11653 12881 y Fx(x)12211 12715 y FA(.)492 b(In)369 b(order)g(to)h(exhibit)g (these)f(n)-31 b(um)g(b)31 b(ers)369 b(within)i(this)f(framew)-31 b(ork)370 b(consider)f(\014rst)7529 16019 y SDict begin H.S end 7529 16019 a 7529 16019 a SDict begin 12 H.A end 7529 16019 a 7529 16019 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3) cvn H.B /DEST pdfmark end 7529 16019 a FB(e)8044 15562 y Fx(c)151 b(x)g(d)9705 15673 y Fp(x)11668 16019 y FA(=)14336 14635 y Fq(1)13978 14967 y Fv(X)13943 17312 y Fx(n)p Fu(=0)16113 15270 y FB(c)16592 14868 y Fx(n)p 16113 15764 1082 45 v 16168 16778 a FB(n)p FA(!)17697 16019 y FB(E)18947 15562 y Fx(n)18514 16293 y(x)20164 16019 y FA(=)615 b(1)247 b(+)23939 14635 y Fq(1)23581 14967 y Fv(X)23547 17312 y Fx(n)p Fu(=1)25716 15270 y FB(c)26195 14868 y Fx(n)p 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23254 y FA(In)300 b(the)h(third)h(step)e(an)h(in)-31 b(terc)g(hange)303 b(of)e(summation)i(has)e(b)31 b(een)300 b(p)31 b(erformed)301 b(\(w)-31 b(e)301 b(ignore)h(questions)g(of)f (con)-31 b(v)g(ergence)302 b(here\),)800 24582 y(and)344 b(in)g(the)h(last)g(step)e(an)i(exp)31 b(onen)-31 b(tial)346 b(generating)g(function)f(\(e.g.f.\))487 b(has)344 b(b)31 b(een)343 b(in)-31 b(tro)31 b(duced)345 b(for)f(the)g FB(m)p Fz(\000)p FA(th)g(column)h(of)800 25910 y(the)372 b(n)-31 b(um)g(b)31 b(er)371 b(triangle,)j(or)d(lo)-31 b(w)g(er)373 b(triangular)g(matrix,)h Fy(S2)p FA(.)499 b(The)372 b(recursion)e(relation)j(implies)g FB(G)p FA(2)42876 26076 y Fx(m)43718 25910 y FA(\()p FB(c)p FA(\))312 b(=)46971 25475 y Fu(1)p 46673 25656 1037 45 v 46673 26292 a Fx(m)p Fu(!)48027 25910 y FA(\()p FB(G)p FA(2\()p FB(c)p FA(\)\))51649 25509 y Fx(m)52493 25910 y FA(,)800 27239 y(with)370 b FB(G)p FA(2\()p FB(c)p FA(\))310 b(=)d FB(exp)p FA(\()p FB(c)p FA(\))247 b Fz(\000)f FA(1;)370 b(therefore)f(w)-31 b(e)370 b(obtain)22912 25910 y SDict begin H.S end 22912 25910 a 22912 25910 a SDict begin 12 H.A end 22912 25910 a 22912 25910 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4) cvn H.B /DEST pdfmark end 22912 25910 a 13591 30561 a FB(e)14106 30104 y Fx(c)152 b(x)f(d)15768 30215 y Fp(x)16931 30561 y FA(=)18919 29177 y Fq(1)18561 29510 y Fv(X)18407 31854 y Fx(m)p Fu(=0)20994 29812 y FA(1)p 20631 30307 1280 45 v 20631 31320 a FB(m)p FA(!)22228 30561 y(\()p FB(G)p FA(2\()p FB(c)p FA(\))184 b FB(x)p FA(\))26667 30104 y Fx(m)27695 30561 y FB(d)28640 30104 y Fx(m)28271 30835 y(x)30096 30561 y FA(=)307 b(:)h FB(e)32394 30104 y Fu(\()p Fx(exp)p Fu(\()p Fx(c)p Fu(\))p Fq(\000)p Fu(1\))p Fx(x)152 b(d)37800 30215 y Fp(x)38656 30561 y FA(:)369 b FB(;)11747 b FA(\(4\))800 34012 y(where)388 b(w)-31 b(e)388 b(ha)-31 b(v)g(e)389 b(used)e(the)h(linear)g(normal)h(order)f (sym)-31 b(b)31 b(ol)388 b(:)339 b FB(A)g FA(:)388 b(from)g(quan)-31 b(tum)390 b(ph)-31 b(ysics.)548 b(\(:)340 b FB(A)e FA(:)388 b(means)h(expand)f FB(A)g FA(in)800 35341 y(p)31 b(o)-31 b(w)g(ers)382 b(of)g FB(x)f FA(and)g FB(d)9508 35507 y Fx(x)10066 35341 y FA(,)k(and)d(mo)-31 b(v)g(e)383 b(all)f(op)31 b(erators)382 b FB(d)22834 35507 y Fx(x)23773 35341 y FA(to)g(the)g(righ)-31 b(t-hand)383 b(side,)i(ignoring)e(the)f (usual)g(comm)-31 b(utation)385 b(rule)800 36669 y([)p FB(d)1683 36835 y Fx(x)2241 36669 y FB(;)184 b(x)p FA(])358 b Fz(\021)e FB(d)5823 36835 y Fx(x)6565 36669 y FB(x)265 b Fz(\000)h FB(x)184 b(d)9983 36835 y Fx(x)10897 36669 y FA(=)356 b(1.)581 b(F)-92 b(or)398 b(example,)408 b(:)357 b(\()p FB(x)184 b(d)22738 36835 y Fx(x)23296 36669 y FA(\))23726 36267 y Fx(m)24924 36669 y FA(:)357 b(=)713 b FB(x)27795 36267 y Fx(m)28820 36669 y FB(d)29396 36267 y Fx(m)29396 36943 y(x)30237 36669 y FB(:)p FA(\))399 b(This)g(normal)h(order)e(prescription)h(is)f(applied)i(to)800 37997 y(eac)-31 b(h)376 b(term)f(of)h(the)f(expanded)h(exp)31 b(onen)-31 b(tial)377 b(in)f(eq.)p 0 .5 0 TeXcolorrgb 23592 37997 a SDict begin H.S end 23592 37997 a 0 .5 0 TeXcolorrgb FA(4)p 0 .5 0 TeXcolorrgb 24146 37284 a SDict begin H.R end 24146 37284 a 24146 37997 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4) cvn H.B /ANN pdfmark end 24146 37997 a Black FA(.)510 b(F)-92 b(rom)375 b(T)-92 b(a)-31 b(ylor's)377 b(theorem,)h(w)-31 b(e)375 b(see)g(that)h(for)f(suitable)i(functions)800 39326 y FB(f)488 b FA(w)-31 b(e)370 b(ha)-31 b(v)g(e)6012 37997 y SDict begin H.S end 6012 37997 a 6012 37997 a SDict begin 12 H.A end 6012 37997 a 6012 37997 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5) cvn H.B /DEST pdfmark end 6012 37997 a 10174 40654 a FB(e)10689 40197 y Fx(c)151 b(x)g(d)12350 40308 y Fp(x)13084 40654 y FB(f)119 b FA(\()p FB(x)p FA(\))615 b(=)308 b(:)g FB(e)18152 40197 y Fu(\()p Fx(exp)p Fu(\()p Fx(c)p Fu(\))p Fq(\000)p Fu(1\))p Fx(x)151 b(d)23557 40308 y Fp(x)24413 40654 y FA(:)493 b FB(f)119 b FA(\()p FB(x)p FA(\))615 b(=)g FB(f)119 b FA(\()p FB(x)247 b FA(+)e(\()p FB(e)33480 40197 y Fx(c)34178 40654 y Fz(\000)h FA(1\))p FB(x)p FA(\))678 b(=)e FB(f)119 b FA(\()p FB(x)41270 40197 y Fq(0)41581 40654 y FA(\))739 b FB(;)8329 b FA(\(5\))800 42647 y(with)440 b FB(x)4024 42245 y Fq(0)4757 42647 y FA(=)423 b FB(e)6556 42245 y Fx(c)7191 42647 y FB(x)p FA(.)700 b(Therefore)439 b(the)f(parameter)i FB(\025)d FA(for)i(\014nite)g(scale)g(transformations)i(is)d FB(\025)422 b FA(=)h FB(e)41825 42245 y Fx(c)42715 42647 y FA(if)439 b FB(c)f FA(is)g(the)h(parameter)800 43975 y(for)367 b(in\014nitesimal)i(transformations.)494 b(In)366 b(the)h(con)-31 b(text)368 b(of)g(Lie)e(groups)h(this)g(fact)g(is)g(found)g(b)-31 b(y)367 b(in)-31 b(tegrating)369 b(the)e(ordinary)800 45303 y(di\013eren)-31 b(tial)371 b(equation)g(\(see)e(for)g(example)i ([)p 0 .5 0 TeXcolorrgb 19415 45303 a SDict begin H.S end 19415 45303 a 0 .5 0 TeXcolorrgb FA(2)p 0 .5 0 TeXcolorrgb 19969 44590 a SDict begin H.R end 19969 44590 a 19969 45303 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Dr) cvn H.B /ANN pdfmark end 19969 45303 a Black FA(])e(and)h(the)f (references)f(giv)-31 b(en)371 b(there\):)36373 43975 y SDict begin H.S end 36373 43975 a 36373 43975 a SDict begin 12 H.A end 36373 43975 a 36373 43975 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark end 36373 43975 a 22775 47628 a FB(dx)p FA(\()p FB(\013)t FA(\))p 22775 48122 2783 45 v 22775 49136 a FB(d\013)26305 48377 y FA(=)615 b FB(c)184 b(x)p FA(\()p FB(\013)t FA(\))p FB(;)20430 b FA(\(6\))800 51463 y(for)378 b(curv)-31 b(es)376 b(starting)j(at)f(a)g(\014xed)f FB(x)321 b FA(:=)h FB(x)p FA(\()p FB(\013)j FA(=)d(0\).)518 b(The)377 b(\014nite)h (transformation)i(maps)e FB(x)f FA(to)h FB(x)41129 51061 y Fq(0)41761 51463 y FA(:=)321 b FB(x)p FA(\()p FB(\013)326 b FA(=)321 b(1\).)518 b FB(x)48970 51061 y Fq(0)49658 51463 y FA(should)800 52791 y(not)377 b(b)31 b(e)375 b(confused)h(with)i(a)e(deriv)-61 b(ativ)-31 b(e.)515 b(In)375 b(this)i(w)-31 b(ork)376 b(w)-31 b(e)377 b(will)h(generalize)f (this)f(to)h(the)f(case)g FB(E)41532 52957 y Fx(k)19 b Fu(;)p Fx(x)43149 52791 y Fz(\021)318 b FB(x)44961 52389 y Fx(k)45690 52791 y FB(d)46266 52957 y Fx(x)47200 52791 y FA(with)377 b FB(k)354 b Fz(2)319 b Fy(Z)p FA(.)800 54120 y(It)369 b(is)g(clear)h(from)g(the)f(solution)i(of)f(the)f (di\013eren)-31 b(tial)371 b(equation)27696 52791 y SDict begin H.S end 27696 52791 a 27696 52791 a SDict begin 12 H.A end 27696 52791 a 27696 52791 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7) cvn H.B /DEST pdfmark end 27696 52791 a 4479 56383 a FB(dx)p FA(\()p FB(\013)t FA(\))p 4479 56877 2783 45 v 4479 57891 a FB(d\013)8009 57132 y FA(=)615 b FB(c)184 b(x)10781 56674 y Fx(k)11326 57132 y FA(\()p FB(\013)t FA(\))p FB(;)924 b FA(with)370 b(initial)i (condition)740 b FB(x)p FA(\()p FB(\013)312 b FA(=)c(0\))677 b(=:)308 b FB(x)738 b FA(and)369 b(its)h(transform)739 b FB(x)42618 56674 y Fq(0)43235 57132 y FA(:=)308 b FB(x)p FA(\()p FB(\013)k FA(=)307 b(1\))p FB(;)2134 b FA(\(7\))800 60154 y(that)330 b(the)e(scale)h(transformation)i(case)d FB(k)342 b FA(=)308 b(1)328 b(whic)-31 b(h)330 b(has)e(b)31 b(een)328 b(treated)h(ab)31 b(o)-31 b(v)g(e)329 b(is)f(sp)31 b(ecial.)480 b(F)-92 b(or)327 b FB(k)343 b Fz(6)p FA(=)307 b(1,)338 b(after)328 b(separation)800 61482 y(of)370 b(v)-61 b(ariables,)370 b(w)-31 b(e)370 b(obtain)h(the)e(equation)i(\() p FB(x)19522 61081 y Fq(0)19833 61482 y FA(\))20263 61081 y Fu(1)p Fq(\000)p Fx(k)22495 61482 y Fz(\000)553 b FB(x)24542 61081 y Fu(1)p Fq(\000)p Fx(k)26834 61482 y FA(=)615 b(\(1)247 b Fz(\000)f FB(k)35 b FA(\))184 b FB(c)p FA(.)494 b(Setting)36965 60154 y SDict begin H.S end 36965 60154 a 36965 60154 a SDict begin 12 H.A end 36965 60154 a 36965 60154 a SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark end 36965 60154 a 21423 63918 a FB(x)22056 63461 y Fq(0)22981 63918 y FA(=)615 b(\(1)247 b(+)f FB(g)40 b FA(\()p FB(k)35 b FA(;)184 b FB(c)p FA(;)g FB(x)p FA(\)\))g FB(x)19212 b FA(\(8\))800 66353 y(w)-31 b(e)370 b(ha)-31 b(v)g(e)4982 65025 y SDict begin H.S end 4982 65025 a 4982 65025 a SDict begin 12 H.A end 4982 65025 a 4982 65025 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark end 4982 65025 a 16458 67681 a FA(1)246 b(+)g FB(g)40 b FA(\()p FB(k)35 b FA(;)184 b FB(c)p FA(;)g FB(x)p FA(\))617 b(=)24590 66785 y Fv(\000)25098 67681 y FA(1)246 b Fz(\000)g FA(\()p FB(k)282 b Fz(\000)246 b FA(1\))184 b FB(c)g(x)31862 67224 y Fx(k)19 b Fq(\000)p Fu(1)33541 66785 y Fv(\001)34048 67014 y Fq(\000)35387 66717 y Fo(1)p 34873 66840 1405 38 v 34873 67281 a Fp(k)9 b Fn(\000)p Fo(1)36835 67681 y FB(:)14244 b FA(\(9\))800 69674 y(Therefore)5818 68345 y SDict begin H.S end 5818 68345 a 5818 68345 a SDict begin 12 H.A end 5818 68345 a 5818 68345 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10) cvn H.B /DEST pdfmark end 5818 68345 a 13349 71002 a FB(e)13864 70545 y Fx(c)151 b(x)14913 70211 y Fp(k)15548 70545 y Fx(d)16009 70656 y Fp(x)16742 71002 y FB(f)119 b FA(\()p FB(x)p FA(\))616 b(=)f FB(f)119 b FA(\()p FB(x)22712 70545 y Fq(0)23023 71002 y FA(\))615 b(=)g FB(f)26390 69773 y Fv(\020)27051 71002 y FA(\(1)247 b Fz(\000)f FA(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(c)g(x)34245 70545 y Fx(k)19 b Fq(\000)p Fu(1)35925 71002 y FA(\))36355 70524 y Fq(\000)37694 70227 y Fo(1)p 37180 70350 1405 38 v 37180 70791 a Fp(k)9 b Fn(\000)p Fo(1)38957 71002 y FB(x)39590 69773 y Fv(\021)50832 71002 y FA(\(10\))p Black 26523 74617 a(2)p Black eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -2672 a SDict begin H.S end 800 -2672 a 0 TeXcolorgray 0 TeXcolorgray 800 -2672 a SDict begin H.R end 800 -2672 a 800 -2672 a SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark end 800 -2672 a Black Black 3875 x FA(for)375 b FB(k)353 b Fz(2)317 b Fy(Z)250 b Fz(n)g(f)p FA(1)p Fz(g)p FA(.)511 b(The)376 b(case)e FB(k)353 b FA(=)317 b(1)375 b(has)g(b)31 b(een)375 b(dealt)h(with)g(in)g(eq.)p 0 .5 0 TeXcolorrgb 29413 1203 a SDict begin H.S end 29413 1203 a 0 .5 0 TeXcolorrgb FA(5)p 0 .5 0 TeXcolorrgb 29967 489 a SDict begin H.R end 29967 489 a 29967 1203 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5) cvn H.B /ANN pdfmark end 29967 1203 a Black FA(.)510 b(It)375 b(can)g(b)31 b(e)374 b(reco)-31 b(v)g(ered)375 b(from)h(eq.)p 0 .5 0 TeXcolorrgb 44819 1203 a SDict begin H.S end 44819 1203 a 0 .5 0 TeXcolorrgb FA(10)p 0 .5 0 TeXcolorrgb 45926 489 a SDict begin H.R end 45926 489 a 45926 1203 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.10) cvn H.B /ANN pdfmark end 45926 1203 a Black 374 w FA(b)-31 b(y)376 b(taking)h(the)800 2531 y(limit)371 b FB(k)282 b Fz(\000)246 b FA(1)307 b Fz(!)h FA(0.)2460 3859 y FB(k)35 b Fz(\000)p FC(Stirling)359 b(numb)-57 b(ers)359 b(of)f(the)f(se)-57 b(c)g(ond)359 b(kind)p FA(,)336 b(whic)-31 b(h)329 b(w)-31 b(e)328 b(will)h(denote)f(b)-31 b(y)328 b FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\),)339 b(with)329 b FB(S)64 b FA(2\(1;)184 b FB(n;)g(m)p FA(\))311 b(=)d FB(S)64 b FA(2\()p FB(n;)184 b(m)p FA(\))800 5188 y(the)427 b(ordinary)g(Stirling)h(subset)e(n)-31 b(um)g(b)31 b(ers,)441 b(emerge)426 b(in)h(a)g(pro)31 b(of,)441 b(indep)31 b(enden)-31 b(t)427 b(of)g(the)g(one)g(implied)h(b)-31 b(y)427 b(eq.)p 0 .5 0 TeXcolorrgb 48090 5188 a SDict begin H.S end 48090 5188 a 0 .5 0 TeXcolorrgb FA(10)p 0 .5 0 TeXcolorrgb 49197 4474 a SDict begin H.R end 49197 4474 a 49197 5188 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.10) cvn H.B /ANN pdfmark end 49197 5188 a Black FA(,)441 b(of)427 b(the)800 6516 y(follo)-31 b(wing)373 b(op)31 b(erator)370 b(iden)-31 b(tit)g(y)-92 b(,)371 b(v)-61 b(alid)371 b(for)e FB(k)343 b Fz(2)307 b Fy(Z)p FA(,)22031 5188 y SDict begin H.S end 22031 5188 a 22031 5188 a SDict begin 12 H.A end 22031 5188 a 22031 5188 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11) cvn H.B /DEST pdfmark end 22031 5188 a 20482 9081 a FB(e)20997 8624 y Fx(c)151 b(x)22046 8290 y Fp(k)22681 8624 y Fx(d)23142 8735 y Fp(x)24306 9081 y FA(=)307 b(:)h FB(e)26604 8624 y Fx(g)28 b Fu(\()p Fx(k)19 b Fu(;)p Fx(c)p Fu(;)p Fx(x)p Fu(\))149 b Fx(x)i(d)30909 8735 y Fp(x)31766 9081 y FA(:)738 b FB(;)17714 b FA(\(11\))800 11474 y(where)416 b FB(g)40 b FA(\()p FB(k)35 b FA(;)184 b FB(c)p FA(;)g FB(x)p FA(\))418 b(is)e(de\014ned)g(b)-31 b(y)416 b(eq.)p 0 .5 0 TeXcolorrgb 17003 11474 a SDict begin H.S end 17003 11474 a 0 .5 0 TeXcolorrgb FA(9)p 0 .5 0 TeXcolorrgb 17556 10760 a SDict begin H.R end 17556 10760 a 17556 11474 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.9) cvn H.B /ANN pdfmark end 17556 11474 a Black 416 w FA(for)g FB(k)421 b Fz(6)p FA(=)385 b(1)417 b(and)f FB(g)40 b FA(\(1;)184 b FB(c)p FA(;)g FB(x)p FA(\))388 b(=)e FB(G)p FA(2\()p FB(c)p FA(\))417 b(\(see)f(eq.)p 0 .5 0 TeXcolorrgb 38081 11474 a SDict begin H.S end 38081 11474 a 0 .5 0 TeXcolorrgb FA(4)p 0 .5 0 TeXcolorrgb 38634 10760 a SDict begin H.R end 38634 10760 a 38634 11474 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4) cvn H.B /ANN pdfmark end 38634 11474 a Black FA(\).)634 b(By)416 b(analogy)i(with)g(eq.)p 0 .5 0 TeXcolorrgb 50293 11474 a SDict begin H.S end 50293 11474 a 0 .5 0 TeXcolorrgb FA(1)p 0 .5 0 TeXcolorrgb 50847 10760 a SDict begin H.R end 50847 10760 a 50847 11474 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 50847 11474 a Black 416 w FA(the)800 12802 y FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(n)-31 b(um)g(b)31 b(er)369 b(triangle)i(is)e(de\014ned)g(b)-31 b(y)21114 11474 y SDict begin H.S end 21114 11474 a 21114 11474 a SDict begin 12 H.A end 21114 11474 a 21114 11474 a SDict begin [ /View [/XYZ H.V] /Dest (equation.12) cvn H.B /DEST pdfmark end 21114 11474 a 9435 16081 a FB(E)11423 15624 y Fx(n)10252 16355 y(k)19 b Fu(;)p Fx(x)12333 16081 y Fz(\021)307 b FA(\()p FB(x)14564 15624 y Fx(k)15293 16081 y FB(d)15869 16247 y Fx(x)16427 16081 y FA(\))16857 15624 y Fx(n)18075 16081 y FA(=)20230 14698 y Fx(n)19705 15030 y Fv(X)19551 17374 y Fx(m)p Fu(=1)21826 16081 y FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))g FB(x)28028 15624 y Fx(m)p Fu(+\()p Fx(k)19 b Fq(\000)p Fu(1\))153 b Fx(n)32749 16081 y FB(d)33694 15624 y Fx(m)33325 16355 y(x)35273 16081 y FB(;)923 b(n)308 b Fz(2)g Fy(N)p FB(;)923 b(k)342 b Fz(2)308 b Fy(Z)369 b FB(;)6667 b FA(\(12\))800 19489 y(with)454 b(the)g(further)e(con)-31 b(v)g(en)g(tion)456 b(that)e FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))451 b(=)d(0)453 b(for)g FB(n)448 b(<)f(m)453 b FA(and)g FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g FA(0\))452 b(=)447 b(0.)745 b(These)453 b(n)-31 b(um)g(b)31 b(ers)452 b(will)j(b)31 b(e)800 20818 y(sho)-31 b(wn)370 b(to)g(satisfy)g(the)f(recursion)g(relation)19683 19489 y SDict begin H.S end 19683 19489 a 19683 19489 a SDict begin 12 H.A end 19683 19489 a 19683 19489 a SDict begin [ /View [/XYZ H.V] /Dest (equation.13) cvn H.B /DEST pdfmark end 19683 19489 a 8584 23210 a FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))619 b(=)c(\(\()p FB(k)282 b Fz(\000)246 b FA(1\)\()p FB(n)h Fz(\000)f FA(1\))h(+)f FB(m)p FA(\))184 b FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n)249 b Fz(\000)d FA(1)p FB(;)184 b(m)p FA(\))556 b(+)d FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n)248 b Fz(\000)e FA(1)p FB(;)184 b(m)247 b Fz(\000)f FA(1\))p FB(;)5818 b FA(\(13\))800 25602 y(with)370 b FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b(1)p FB(;)g FA(1\))312 b(=)307 b(1)370 b(from)g(eq.)p 0 .5 0 TeXcolorrgb 14948 25602 a SDict begin H.S end 14948 25602 a 0 .5 0 TeXcolorrgb FA(12)p 0 .5 0 TeXcolorrgb 16055 24889 a SDict begin H.R end 16055 24889 a 16055 25602 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.12) cvn H.B /ANN pdfmark end 16055 25602 a Black FA(.)493 b(The)369 b(e.g.f.)494 b(for)370 b(the)f FB(m)p Fz(\000)p FA(th)g(column)i(of)f(the)f Fy(S2)q FA(\()p FB(k)35 b FA(\))370 b(triangle,)43372 24274 y SDict begin H.S end 43372 24274 a 43372 24274 a SDict begin 12 H.A end 43372 24274 a 43372 24274 a SDict begin [ /View [/XYZ H.V] /Dest (equation.14) cvn H.B /DEST pdfmark end 43372 24274 a 17793 28881 a FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(m)p FA(;)g FB(x)p FA(\))618 b(:=)26241 27498 y Fq(1)25882 27830 y Fv(X)25676 30154 y Fx(n)p Fu(=)p Fx(m)28057 28881 y FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))33763 28133 y FB(x)34396 27731 y Fx(n)p 33763 28627 1236 45 v 33894 29641 a FB(n)p FA(!)35500 28881 y FB(;)15025 b FA(\(14\))800 32208 y(satis\014es)4877 30880 y SDict begin H.S end 4877 30880 a 4877 30880 a SDict begin 12 H.A end 4877 30880 a 4877 30880 a SDict begin [ /View [/XYZ H.V] /Dest (equation.15) cvn H.B /DEST pdfmark end 4877 30880 a 19286 33781 a FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(m)p FA(;)g FB(x)p FA(\))619 b(=)27358 33032 y(1)p 26995 33527 1280 45 v 26995 34541 a FB(m)p FA(!)28591 33781 y(\()p FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\)\))33469 33324 y Fx(m)50832 33781 y FA(\(15\))800 36058 y(for)369 b FB(k)343 b Fz(6)p FA(=)307 b(1,)370 b(with)8333 34730 y SDict begin H.S end 8333 34730 a 8333 34730 a SDict begin 12 H.A end 8333 34730 a 8333 34730 a SDict begin [ /View [/XYZ H.V] /Dest (equation.16) cvn H.B /DEST pdfmark end 8333 34730 a 18598 37610 a FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))618 b(=)d(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(g)40 b FA(2\()p FB(k)35 b FA(;)32003 36861 y FB(x)p 31060 37355 2518 45 v 31060 38369 a(k)281 b Fz(\000)246 b FA(1)33711 37610 y(\))369 b FB(;)16015 b FA(\(16\))800 39979 y(where)4001 38651 y SDict begin H.S end 4001 38651 a 4001 38651 a SDict begin 12 H.A end 4001 38651 a 4001 38651 a SDict begin [ /View [/XYZ H.V] /Dest (equation.17) cvn H.B /DEST pdfmark end 4001 38651 a 19691 41918 a FB(g)40 b FA(2\()p FB(k)35 b FA(;)184 b FB(y)40 b FA(\))616 b(:=)26149 40534 y Fq(1)25791 40866 y Fv(X)25757 43210 y Fx(n)p Fu(=1)27794 41918 y FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g FA(1\))g FB(y)33303 41461 y Fx(n)50832 41918 y FA(\(17\))800 44900 y(is)369 b(the)g(ordinary)g(generating)i(function)f(\(o.g.f.\))496 b(for)369 b(the)g(\014rst)f(column)i(of)g(the)f(triangle)h(of)g(n)-31 b(um)g(b)31 b(ers)368 b FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(whic)-31 b(h)800 46229 y(is)369 b(asso)31 b(ciated)371 b(to)e(triangle)i FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(b)-31 b(y)p 0 .5 0 TeXcolorrgb 19524 46229 a SDict begin H.S end 19524 46229 a -402 x Fu(2)20021 46229 y SDict begin 12 H.L end 20021 46229 a 20021 46229 a SDict begin [ /Subtype /Link /Dest (Hfootnote.2) cvn /H /I /Border [0 0 0] /Color [1 0 0] H.B /ANN pdfmark end 20021 46229 a Black 20390 44900 a SDict begin H.S end 20390 44900 a 20390 44900 a SDict begin 12 H.A end 20390 44900 a 20390 44900 a SDict begin [ /View [/XYZ H.V] /Dest (equation.18) cvn H.B /DEST pdfmark end 20390 44900 a 16300 49198 a FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))618 b(:=)e(\()p FB(k)282 b Fz(\000)245 b FA(1\))27240 48740 y Fx(n)p Fq(\000)p Fx(m)29639 48449 y FB(m)p FA(!)p 29639 48943 1280 45 v 29793 49957 a FB(n)p FA(!)31235 49198 y FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b FB(:)13532 b FA(\(18\))800 51962 y(These)318 b(n)-31 b(um)g(b)31 b(er)319 b(triangles,)330 b(or)318 b(lo)-31 b(w)g(er)320 b(triangular)g(in\014nite-dimensional)i (matrices,)330 b Fy(s2)q FA(\()p FB(k)35 b FA(\),)330 b(whic)-31 b(h)319 b(are)f(here)g(only)h(de\014ned)800 53290 y(for)369 b FB(k)343 b Fz(2)307 b Fy(Z)246 b Fz(n)h(f)p FA(1)p Fz(g)p FA(,)371 b(ob)31 b(ey)369 b(the)h(recursion)e(relation) 21979 51962 y SDict begin H.S end 21979 51962 a 21979 51962 a SDict begin 12 H.A end 21979 51962 a 21979 51962 a SDict begin [ /View [/XYZ H.V] /Dest (equation.19) cvn H.B /DEST pdfmark end 21979 51962 a 6294 56259 a FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))619 b(=)13683 55510 y FB(k)281 b Fz(\000)246 b FA(1)p 13683 56004 2518 45 v 14609 57018 a FB(n)16702 56259 y FA([)q(\()p FB(k)281 b Fz(\000)246 b FA(1\)\()p FB(n)i Fz(\000)e FA(1\))h(+)f FB(m)p FA(])369 b FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n)249 b Fz(\000)d FA(1)p FB(;)184 b(m)p FA(\))555 b(+)35993 55510 y FB(m)p 35993 56004 972 45 v 36147 57018 a(n)37283 56259 y(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n)248 b Fz(\000)e FA(1)p FB(;)184 b(m)247 b Fz(\000)f FA(1\))739 b FB(;)3527 b FA(\(19\))800 58961 y(with)3321 57633 y SDict begin H.S end 3321 57633 a 3321 57633 a SDict begin 12 H.A end 3321 57633 a 3321 57633 a SDict begin [ /View [/XYZ H.V] /Dest (equation.20) cvn H.B /DEST pdfmark end 3321 57633 a 11944 60290 a FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))619 b(=)c(0)p FB(;)554 b(n)308 b(<)f(m)738 b(;)923 b(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g FA(0\))620 b(=)614 b(0)p FB(;)924 b(s)p FA(2\()p FB(k)35 b FA(;)184 b(1)p FB(;)g FA(1\))312 b(=)307 b(1)370 b FB(:)9176 b FA(\(20\))800 62256 y(It)324 b(follo)-31 b(ws)326 b(that)f(these)f(n) -31 b(um)g(b)31 b(ers)323 b(are)h(nonnegativ)-31 b(e.)480 b(A)-31 b(t)324 b(this)g(stage)h(it)f(is)g(not)h(ob)-31 b(vious)325 b(that)g(they)f(are)g(in)-31 b(tegers)324 b(for)g(ev)-31 b(ery)800 63585 y FB(k)343 b Fz(2)307 b Fy(Z)246 b Fz(n)h(f)p FA(1)p Fz(g)p FA(.)800 64913 y(The)369 b(o.g.f.)495 b(of)370 b(the)f FB(m)p FA(-th)h(column)g(of)g (the)f Fy(s2)q FA(\()p FB(k)35 b FA(\))370 b(matrix)h(is)26593 63585 y SDict begin H.S end 26593 63585 a 26593 63585 a SDict begin 12 H.A end 26593 63585 a 26593 63585 a SDict begin [ /View [/XYZ H.V] /Dest (equation.21) cvn H.B /DEST pdfmark end 26593 63585 a 14509 68192 a FB(g)40 b FA(2\()p FB(k)35 b FA(;)184 b FB(m)p FA(;)g FB(y)40 b FA(\))617 b(:=)22604 66809 y Fq(1)22245 67141 y Fv(X)22039 69465 y Fx(n)p Fu(=)p Fx(m)24420 68192 y FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))g FB(y)30348 67735 y Fx(n)31569 68192 y FA(=)615 b(\()p FB(g)40 b FA(2\()p FB(k)35 b FA(;)184 b FB(y)40 b FA(\)\))37571 67735 y Fx(m)38784 68192 y FB(;)11741 b FA(\(21\))p Black 800 70267 20800 45 v 2032 70983 a Ft(2)p 0 TeXcolorgray 2494 70244 a SDict begin H.S end 2494 70244 a 0 TeXcolorgray 0 TeXcolorgray 2494 70244 a SDict begin H.R end 2494 70244 a 2494 70244 a SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.2) cvn H.B /DEST pdfmark end 2494 70244 a Black 1052 x Fs(These)313 b Fm(asso)-49 b(ciate)g(d)340 b(Stirling)f(numb)-49 b(ers)342 b(of)d(the)h(se)-49 b(c)g(ond)341 b(kind)314 b Fs(are)g(not)g(the)f(ones)h(of)g([)p 0 .5 0 TeXcolorrgb 32891 71296 a SDict begin H.S end 32891 71296 a 0 .5 0 TeXcolorrgb Fs(5)p 0 .5 0 TeXcolorrgb 33362 70725 a SDict begin H.R end 33362 70725 a 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/Dest (cite.Roman) cvn H.B /ANN pdfmark end 28648 31352 a Black FA(]\))416 b(the)e(p)31 b(olynomials)417 b FB(S)64 b FA(2)39249 31518 y Fx(n)39852 31352 y FA(\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))416 b(are)e(a)h(sp)31 b(ecial)415 b(t)-31 b(yp)31 b(e)414 b(of)800 32680 y FC(She\013er)285 b FA(p)31 b(olynomials)288 b(called)e FC(asso)-57 b(ciate)g(d)319 b(p)-57 b(olynomial)319 b(se)-57 b(quenc)g(es)p FA(.)466 b(An)285 b(equiv)-61 b(alen)-31 b(t)287 b(notation)h(used)c(there)h (for)g(the)g(general)800 34008 y(case)380 b(is)h(\\She\013er)f(for)h (\(1)p FB(;)184 b(f)119 b FA(\()p FB(t)p FA(\)\)".)530 b(In)380 b(our)h(case)f FB(f)119 b FA(\()p FB(t)p FA(\))654 b(=)p 25248 33045 3788 45 v 653 w FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(t)p FA(\))r(\),)384 b(where)p 33369 33045 V 381 w FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(t)p FA(\))655 b(=)e(\()p Fz(\000)p FA(1)255 b(+)e(\(1)i(+)e FB(t)p FA(\))45718 33607 y Fu(1)p Fq(\000)p Fx(k)47396 34008 y FA(\))p FB(=)p FA(\(1)j Fz(\000)d FB(k)35 b FA(\))381 b(if)800 35337 y FB(k)h Fz(6)p FA(=)346 b(1.)563 b(This)393 b(is)f(the)h(comp)31 b(ositional)396 b(in)-31 b(v)g(erse)393 b(of)g FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(t)p FA(\))395 b(from)e(eq.)p 0 .5 0 TeXcolorrgb 30648 35337 a SDict begin H.S end 30648 35337 a 0 .5 0 TeXcolorrgb FA(24)p 0 .5 0 TeXcolorrgb 31755 34623 a SDict begin H.R end 31755 34623 a 31755 35337 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.24) cvn H.B /ANN pdfmark end 31755 35337 a Black FA(.)562 b(Also)p 35145 34374 3730 45 v 394 w FB(G)p FA(2\(1;)184 b FB(t)p FA(\))349 b(=)d FB(l)22 b(n)p FA(\(1)263 b(+)f FB(t)p FA(\))392 b(can)h(b)31 b(e)392 b(obtained)800 36665 y(in)370 b(the)f(limit)i(as)e(1)247 b Fz(\000)f FB(k)342 b Fz(!)308 b FA(0.)800 37993 y(F)-92 b(or)455 b(negativ)-31 b(e)458 b FB(k)490 b FA(the)456 b Fy(S2)q FA(\()p FB(k)35 b FA(\))457 b(matrices)f(also)h(con)-31 b(tain)457 b(negativ)-31 b(e)458 b(en)-31 b(tries.)753 b(The)456 b(recursion)f(of)h(eq.)p 0 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Fz(\000)184 b FB(G)p FA(2\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b Fz(\000)p FB(x)p FA(\))619 b(=)c(1)554 b Fz(\000)g FA(\(1)247 b Fz(\000)e FA(\()p Fz(j)p FB(k)35 b Fz(j)248 b FA(+)e(1\))184 b FB(x)p FA(\))40207 50390 y Fo(1)p 39454 50513 1882 38 v 39454 50966 a Fn(j)p Fp(k)9 b Fn(j)p Fo(+1)41524 51223 y FB(:)9001 b FA(\(29\))800 53216 y(Eqs.)p 0 .5 0 TeXcolorrgb 3234 53216 a SDict begin H.S end 3234 53216 a 0 .5 0 TeXcolorrgb FA(19)p 0 .5 0 TeXcolorrgb 4341 52502 a SDict begin H.R end 4341 52502 a 4341 53216 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.19) cvn H.B /ANN pdfmark end 4341 53216 a Black 353 w FA(and)p 0 .5 0 TeXcolorrgb 7183 53216 a SDict begin H.S end 7183 53216 a 0 .5 0 TeXcolorrgb FA(20)p 0 .5 0 TeXcolorrgb 8290 52502 a SDict begin H.R end 8290 52502 a 8290 53216 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.20) cvn H.B /ANN pdfmark end 8290 53216 a Black 353 w FA(will)354 b(b)31 b(e)353 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FA(10)p 0 .5 0 TeXcolorrgb 26733 55159 a SDict begin H.R end 26733 55159 a 26733 55872 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Sloane) cvn H.B /ANN pdfmark end 26733 55872 a Black FA(]\).)839 b(This)485 b(data-base)h(will)g (henceforth)f(b)31 b(e)484 b(quoted)i(as)e FC(EIS)800 57201 y FA(\(Encyclop)31 b(edia)386 b(of)e(In)-31 b(teger)384 b(Sequences\).)535 b(These)384 b(tables)g(also)h(giv)-31 b(e)384 b(the)g FB(A)p FA(-n)-31 b(um)g(b)31 b(ers)383 b(of)h(the)g(sequences)f(formed)h(b)-31 b(y)384 b(the)800 58529 y(\014rst)369 b(columns)h(of)f(the)h(lo)-31 b(w)g(er)370 b(triangular)h(matrices,)g(and)e(of)h(the)f(sequences)g(of)g(the)h(ro) -31 b(w)370 b(sums)e(of)i(these)f(matrices.)800 59857 y(F)-92 b(or)379 b FB(l)348 b FA(=)325 b(2)381 b(the)f(function)h FB(c)p FA(2\()p FB(l)22 b FA(;)184 b FB(y)40 b FA(\))382 b(de\014ned)e(in)g(eq.)p 0 .5 0 TeXcolorrgb 22447 59857 a SDict begin H.S end 22447 59857 a 0 .5 0 TeXcolorrgb FA(23)p 0 .5 0 TeXcolorrgb 23554 59144 a SDict begin H.R end 23554 59144 a 23554 59857 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.23) cvn H.B /ANN pdfmark end 23554 59857 a Black 380 w FA(generates)h(the)f(w)-31 b(ell-kno)g(wn)383 b FC(Catalan)c FA(n)-31 b(um)g(b)31 b(ers.)525 b(F)-92 b(or)379 b FB(l)347 b Fz(2)326 b Fy(Z)253 b Fz(n)h(f)p FA(0)p Fz(g)800 61186 y FA(it)446 b(de\014nes)e(what)i(w)-31 b(e)445 b(call)h FB(l)22 b Fz(\000)p FC(Catalan)445 b FA(n)-31 b(um)g(b)31 b(ers.)719 b(F)-92 b(or)444 b(p)31 b(ositiv)-31 b(e)447 b FB(l)466 b FA(these)445 b(sequences)f(w)-31 b(ere)445 b(in)-31 b(tro)31 b(duced)446 b(b)-31 b(y)445 b(O.)g(Gerard)800 62514 y(in)c FC(EIS)p FA(,)h(who)f(called)h(them)g FC(Patalan)e FA(n)-31 b(um)g(b)31 b(ers.)708 b(It)441 b(will)h(b)31 b(e)441 b(pro)-31 b(v)g(ed)441 b(later)h(\(see)f(Note)h (11\))g(that)g FB(c)p FA(2\()p FB(l)22 b FA(;)184 b FB(y)40 b FA(\))443 b(do)31 b(es)441 b(indeed)800 63842 y(generate)370 b(in)-31 b(tegers.)493 b(Their)370 b(explicit)h(form)f(can)f(b)31 b(e)369 b(found)g(in)h(eq.)p 0 .5 0 TeXcolorrgb 28985 63842 a SDict begin H.S end 28985 63842 a 0 .5 0 TeXcolorrgb FA(77)p 0 .5 0 TeXcolorrgb 30091 63129 a SDict begin H.R end 30091 63129 a 30091 63842 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.77) cvn H.B /ANN pdfmark end 30091 63842 a Black FA(.)2460 65325 y(The)498 b FC(Stirling)516 b(numb)-57 b(ers)515 b(of)f(the)g(\014rst)f(kind)p FA(,)531 b FB(S)64 b FA(1\()p FB(n;)184 b(m)p FA(\))500 b(\()p FB(S)28075 64750 y Fu(\()p Fx(m)p Fu(\))28011 65454 y Fx(n)30105 65325 y FA(in)e([)p 0 .5 0 TeXcolorrgb 31832 65325 a SDict begin H.S end 31832 65325 a 0 .5 0 TeXcolorrgb FA(1)p 0 .5 0 TeXcolorrgb 32386 64611 a SDict begin H.R end 32386 64611 a 32386 65325 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.AS) cvn H.B /ANN pdfmark end 32386 65325 a Black FA(],)530 b FC(EIS)p FA(:)p 0 1 0 0 TeXcolorcmyk 36135 65546 a SDict begin H.S end 36135 65546 a 0 1 0 0 TeXcolorcmyk -221 x FA(A008275)p 36135 65502 4152 45 v 0 1 0 0 TeXcolorcmyk 40286 64568 a SDict begin H.R end 40286 64568 a 40286 65325 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A008275) >> /Subtype /Link H.B /ANN pdfmark end 40286 65325 a Black 497 w FA(can)498 b(b)31 b(e)497 b(de\014ned)h(from)g(the)800 66653 y(in)-31 b(v)g(ersion)370 b(of)g(Eq.)p 0 .5 0 TeXcolorrgb 8727 66653 a SDict begin H.S end 8727 66653 a 0 .5 0 TeXcolorrgb FA(1)p 0 .5 0 TeXcolorrgb 9281 65940 a SDict begin H.R end 9281 65940 a 9281 66653 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 9281 66653 a Black 369 w FA(b)-31 b(y)11187 65325 y SDict begin H.S end 11187 65325 a 11187 65325 a SDict begin 12 H.A end 11187 65325 a 11187 65325 a SDict begin [ /View [/XYZ H.V] /Dest (equation.30) cvn H.B /DEST pdfmark end 11187 65325 a 18457 68807 a FB(x)19090 68350 y Fx(n)19877 68807 y FB(d)20822 68350 y Fx(n)20453 69081 y(x)22039 68807 y FA(=)24195 67423 y Fx(n)23669 67755 y Fv(X)23515 70100 y Fx(m)p Fu(=1)25606 68807 y FB(S)64 b FA(1\()p FB(n;)184 b(m)p FA(\))g(\()p FB(x)g(d)31896 68973 y Fx(x)32457 68807 y FA(\))32887 68350 y Fx(m)34835 68807 y FB(:)15690 b FA(\(30\))p Black 26523 74617 a(4)p Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -2672 a SDict begin H.S end 800 -2672 a 0 TeXcolorgray 0 TeXcolorgray 800 -2672 a SDict begin H.R end 800 -2672 a 800 -2672 a SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark end 800 -2672 a Black Black 3875 x FA(W)-92 b(e)437 b(also)i(set)f FB(S)64 b FA(1\()p FB(n;)184 b FA(0\))425 b Fz(\021)e FA(0)438 b(and)g FB(S)64 b FA(1\()p FB(n;)184 b(m)p FA(\))425 b(:=)d(0)439 b(for)f FB(n)422 b(<)h(m)p FA(.)699 b(In)437 b(\(in\014nite-dimensional\))443 b(matrix)c(notation)i(w)-31 b(e)439 b(can)800 2531 y(write)370 b([)p 0 .5 0 TeXcolorrgb 3939 2531 a SDict begin H.S end 3939 2531 a 0 .5 0 TeXcolorrgb FA(4)p 0 .5 0 TeXcolorrgb 4493 1818 a SDict begin H.R end 4493 1818 a 4493 2531 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Knuth) cvn H.B /ANN pdfmark end 4493 2531 a Black FA(])5169 1203 y SDict begin H.S end 5169 1203 a 5169 1203 a SDict begin 12 H.A end 5169 1203 a 5169 1203 a SDict begin [ /View [/XYZ H.V] /Dest (equation.31) cvn H.B /DEST pdfmark end 5169 1203 a 20228 3859 a Fy(S1)246 b Fz(\001)h Fy(S2)615 b FA(:=)h Fy(1)f FA(=)g Fy(S2)247 b Fz(\001)f Fy(S1)739 b FB(:)17460 b FA(\(31\))800 6487 y(The)443 b FC(signless)464 b(Stirling)i(numb)-57 b(ers)465 b(of)f(the)g(\014rst)f(kind)p FA(,)g(also)444 b(kno)-31 b(wn)444 b(as)f FC(cycle)465 b(numb)-57 b(ers)p FA(,)463 b FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b(m)p FA(\))445 b(\(or)46207 4926 y Fv(\024)47498 5812 y FB(n)47345 7140 y(m)48870 4926 y Fv(\025)49897 6487 y FA(in)f(the)800 8480 y(notation)372 b(of)e([)p 0 .5 0 TeXcolorrgb 6796 8480 a SDict begin H.S end 6796 8480 a 0 .5 0 TeXcolorrgb FA(3)p 0 .5 0 TeXcolorrgb 7350 7766 a SDict begin H.R end 7350 7766 a 7350 8480 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.GKP) cvn H.B /ANN pdfmark end 7350 8480 a Black FA(]\),)g(are)10612 7151 y SDict begin H.S end 10612 7151 a 10612 7151 a SDict begin 12 H.A end 10612 7151 a 10612 7151 a SDict begin [ /View [/XYZ H.V] /Dest (equation.32) cvn H.B /DEST pdfmark end 10612 7151 a 18244 9808 a FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b(m)p FA(\))618 b(:=)d(\()p Fz(\000)p FA(1\))27759 9351 y Fx(n)p Fq(\000)p Fx(m)30025 9808 y FB(S)64 b FA(1\()p FB(n;)184 b(m)p FA(\))740 b FB(:)15477 b FA(\(32\))800 11771 y(Their)370 b(recurrence)d(form)-31 b(ula)371 b(is)14351 10443 y SDict begin H.S end 14351 10443 a 14351 10443 a SDict begin 12 H.A end 14351 10443 a 14351 10443 a SDict begin [ /View [/XYZ H.V] /Dest (equation.33) cvn H.B /DEST pdfmark end 14351 10443 a 14654 14158 a FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n)246 b FA(+)g(1)p FB(;)184 b(m)p FA(\))617 b(=)e FB(n)184 b(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b(m)p FA(\))557 b(+)c FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b(m)248 b Fz(\000)e FA(1\))739 b FB(;)11886 b FA(\(33\))800 16545 y(with)370 b FB(S)64 b FA(1)p FB(p)p FA(\(1)p FB(;)184 b FA(1\))311 b(=)c(1,)370 b FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b FA(0\))311 b(=)c(0)370 b(and)f FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b(m)p FA(\))310 b(=)d(0)370 b(for)f FB(n)308 b(<)f(m)p FA(.)2460 17874 y(The)495 b FC(gener)-57 b(alize)g(d)495 b FB(k)35 b Fz(\000)p FC(Stirling)512 b(numb)-57 b(ers)512 b(of)f(the)f(\014rst)g(kind)495 b FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))497 b(are)d(de\014ned)g(analogously)j(b)-31 b(y)494 b(in)-31 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FA(\))32291 20899 y Fx(m)34054 21356 y FA(for)739 b FB(k)342 b Fz(2)308 b Fy(Z)369 b FB(;)554 b(n)308 b Fz(2)f Fy(N)369 b FB(:)7053 b FA(\(34\))800 24261 y(In)369 b(matrix)h (notation:)10643 22933 y SDict begin H.S end 10643 22933 a 10643 22933 a SDict begin 12 H.A end 10643 22933 a 10643 22933 a SDict begin [ /View [/XYZ H.V] /Dest (equation.35) cvn H.B /DEST pdfmark end 10643 22933 a 14850 25590 a Fy(S1)q FA(\()p FB(k)35 b FA(\))247 b Fz(\001)f Fy(S2)q FA(\()p FB(k)35 b FA(\))616 b(=)f Fy(1)g FA(=)g Fy(S2)q FA(\()p FB(k)35 b FA(\))247 b Fz(\001)g Fy(S1)p FA(\()p FB(k)35 b FA(\))1108 b(for)739 b FB(k)342 b Fz(2)308 b Fy(Z)p FB(:)12082 b FA(\(35\))800 27553 y(F)-92 b(or)368 b FB(k)343 b Fz(2)308 b Fy(N)369 b FA(w)-31 b(e)369 b(de\014ne)g(the)h FC(nonne)-57 b(gative)371 b FB(k)35 b Fz(\000)p FC(Stirling)397 b(numb)-57 b(ers)397 b(of)f(the)g(\014rst)g(kind)369 b FA(b)-31 b(y)38111 26225 y SDict begin H.S end 38111 26225 a 38111 26225 a SDict begin 12 H.A end 38111 26225 a 38111 26225 a SDict begin [ /View [/XYZ H.V] /Dest (equation.36) cvn H.B /DEST pdfmark end 38111 26225 a 17326 29940 a FB(S)64 b FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))618 b(:=)e(\()p Fz(\000)p FA(1\))27944 29483 y Fx(n)p Fq(\000)p Fx(m)30209 29940 y FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b FB(:)14558 b FA(\(36\))800 32327 y(F)-92 b(or)368 b Fz(\000)p FB(k)343 b Fz(2)307 b Fy(N)6607 32493 y Fu(0)7473 32327 y FA(the)370 b(n)-31 b(um)g(b)31 b(ers)369 b FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))372 b(are)d(nonnegativ)-31 b(e.)800 33655 y(The)369 b(recurrence)f (relation)j(for)e FB(k)35 b Fz(\000)p FA(Stirling)371 b(n)-31 b(um)g(b)31 b(ers)369 b(of)h(the)f(\014rst)g(kind)h(is)33126 32327 y SDict begin H.S end 33126 32327 a 33126 32327 a SDict begin 12 H.A end 33126 32327 a 33126 32327 a SDict begin [ /View [/XYZ H.V] /Dest (equation.37) cvn H.B /DEST pdfmark end 33126 32327 a 8246 36042 a FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))618 b(=)d Fz(\000)p FA([\()p FB(k)282 b Fz(\000)246 b FA(1\))184 b FB(m)248 b FA(+)e FB(n)g Fz(\000)g FA(1])184 b FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n)250 b Fz(\000)c FA(1)p FB(;)184 b(m)p FA(\))555 b(+)e FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n)248 b Fz(\000)e FA(1)p FB(;)184 b(m)248 b Fz(\000)e FA(1\))739 b FB(;)5478 b FA(\(37\))800 38429 y(with)490 b FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b(1)p FB(;)g FA(1\))511 b(=)506 b(1,)520 b FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g FA(0\))510 b(=)d(0)489 b(and)g FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))510 b(=)c(0)489 b(for)g FB(n)507 b(<)g(m)p FA(.)850 b(F)-92 b(or)488 b FB(k)542 b Fz(6)p FA(=)507 b(1)489 b(w)-31 b(e)489 b(also)g(in)-31 b(tro)31 b(duce)490 b(the)800 39757 y FC(asso)-57 b(ciate)g(d)396 b FB(k)35 b Fz(\000)p FC(Stirling)397 b(numb)-57 b(ers)397 b(of)f(the)g(\014rst)g(kind)p 0 .5 0 TeXcolorrgb 23646 39757 a SDict begin H.S end 23646 39757 a -401 x Fu(3)24142 39757 y SDict begin 12 H.L end 24142 39757 a 24142 39757 a SDict begin [ /Subtype /Link /Dest (Hfootnote.3) cvn /H /I /Border [0 0 0] /Color [1 0 0] H.B /ANN pdfmark end 24142 39757 a Black 24511 38429 a SDict begin H.S end 24511 38429 a 24511 38429 a SDict begin 12 H.A end 24511 38429 a 24511 38429 a SDict begin [ /View [/XYZ H.V] /Dest (equation.38) cvn H.B /DEST pdfmark end 24511 38429 a 16300 42660 a FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))618 b(:=)e(\(1)247 b Fz(\000)f FB(k)35 b FA(\))27241 42202 y Fx(n)p Fq(\000)p Fx(m)29639 41911 y FB(m)p FA(!)p 29639 42405 1280 45 v 29793 43419 a FB(n)p FA(!)31235 42660 y FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b FB(;)13532 b FA(\(38\))800 45357 y(whic)-31 b(h)370 b(turn)f(out)h(to)g(b)31 b(e)369 b(alw)-31 b(a)g(ys)371 b(nonnegativ)-31 b(e.)495 b(They)370 b(satisfy)g(the)g(recursion)34350 44028 y SDict begin H.S end 34350 44028 a 34350 44028 a SDict begin 12 H.A end 34350 44028 a 34350 44028 a SDict begin [ /View [/XYZ H.V] /Dest (equation.39) cvn H.B /DEST pdfmark end 34350 44028 a 7447 48259 a FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))619 b(=)14836 47510 y FB(k)281 b Fz(\000)246 b FA(1)p 14836 48004 2518 45 v 15763 49018 a FB(n)17486 47362 y Fv(\000)17994 48259 y FA(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(m)248 b FA(+)e FB(n)g Fz(\000)g FA(1)26452 47362 y Fv(\001)27144 48259 y FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n)249 b Fz(\000)d FA(1)p FB(;)184 b(m)p FA(\))248 b(+)35701 47510 y FB(m)p 35701 48004 972 45 v 35855 49018 a(n)36991 48259 y(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n)248 b Fz(\000)e FA(1)p FB(;)184 b(m)247 b Fz(\000)f FA(1\))4865 b(\(39\))800 51018 y(with)392 b FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))346 b(=)c(0)391 b(for)f FB(n)343 b(<)g(m)p FA(,)395 b FB(s)p FA(1\()p FB(k)35 b FA(;)184 b(1)p FB(;)g FA(1\))347 b(=)c(1)390 b(and)h FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g FA(0\))347 b(:=)c(0.)556 b(A)-31 b(t)391 b(this)f(stage)h(it)g(is)g(not)g(ob)-31 b(vious)391 b(that)h(the)800 52346 y FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(are)c(in)g(fact)h(in)-31 b(tegers.)800 53674 y(The)369 b(o.g.f.)495 b(for)369 b(the)h FB(m)p Fz(\000)p FA(th)f(column)i(of)e(the)h(n)-31 b(um)g(b)31 b(er)369 b(triangle)i FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(will)e(b)31 b(e)368 b(sho)-31 b(wn)370 b(to)g(b)31 b(e)43267 52346 y SDict begin H.S end 43267 52346 a 43267 52346 a SDict begin 12 H.A end 43267 52346 a 43267 52346 a SDict begin [ /View [/XYZ H.V] /Dest (equation.40) cvn H.B /DEST pdfmark end 43267 52346 a 14663 56948 a FB(g)40 b FA(1\()p FB(k)35 b FA(;)184 b FB(m)p FA(;)g FB(y)40 b FA(\))617 b(=)22450 55565 y Fq(1)22092 55897 y Fv(X)21885 58221 y Fx(n)p Fu(=)p Fx(m)24266 56948 y FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))g FB(y)30194 56491 y Fx(n)31416 56948 y FA(=)615 b(\()p FB(g)40 b FA(1\()p FB(k)35 b FA(;)184 b FB(y)40 b FA(\)\))37418 56491 y Fx(m)38630 56948 y FB(;)11895 b FA(\(40\))800 60331 y(for)369 b FB(k)343 b Fz(2)307 b Fy(Z)246 b Fz(n)h(f)p FA(1)p Fz(g)p FA(,)371 b(with)11140 59003 y SDict begin H.S end 11140 59003 a 11140 59003 a SDict begin 12 H.A end 11140 59003 a 11140 59003 a SDict begin [ /View [/XYZ H.V] /Dest (equation.41) cvn H.B /DEST pdfmark end 11140 59003 a 20641 61660 a FB(g)40 b FA(1\()p FB(k)35 b FA(;)184 b FB(y)40 b FA(\))616 b(=)f FB(y)224 b(c)p FA(1\()p FB(k)282 b Fz(\000)246 b FA(1;)184 b FB(y)40 b FA(\))p FB(;)17875 b FA(\(41\))800 63623 y(and,)370 b(for)f FB(l)330 b Fz(2)307 b Fy(Z)246 b Fz(n)h(f)p FA(0)p Fz(g)p FA(,)10820 62295 y SDict begin H.S end 10820 62295 a 10820 62295 a SDict begin 12 H.A end 10820 62295 a 10820 62295 a SDict begin [ /View [/XYZ H.V] /Dest (equation.42) cvn H.B /DEST pdfmark end 10820 62295 a 19473 65699 a FB(c)p FA(1\()p FB(l)22 b FA(;)184 b FB(y)40 b FA(\))617 b(=)25017 64950 y Fz(\000)p FA(1)554 b(+)f(\(1)247 b Fz(\000)f FB(l)207 b(y)40 b FA(\))32286 64549 y Fq(\000)p Fx(l)p 25017 65445 8301 45 v 28359 66459 a FB(l)28711 66139 y Fu(2)29393 66459 y FB(y)33819 65699 y(:)16706 b FA(\(42\))800 68250 y(Hence)502 b Fy(s1)q FA(\()p FB(k)35 b FA(\))503 b(is,)537 b(lik)-31 b(e)503 b Fy(s2)q FA(\()p FB(k)35 b FA(\),)537 b(a)503 b(con)-31 b(v)g(olution)506 b(triangle)e(generated)f(from)g(its)g FB(m)530 b FA(=)f(1)503 b(column,)537 b FC(i.e.)894 b FA(b)31 b(oth)503 b(are)f(Bell)800 69578 y(matrices.)p Black 800 70267 20800 45 v 2032 70983 a Ft(3)p 0 TeXcolorgray 2494 70244 a SDict begin H.S end 2494 70244 a 0 TeXcolorgray 0 TeXcolorgray 2494 70244 a SDict begin H.R end 2494 70244 a 2494 70244 a SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.3) cvn H.B /DEST pdfmark end 2494 70244 a Black 1052 x Fs(These)313 b(asso)26 b(ciated)316 b(Stirling)e(n)-26 b(um)g(b)26 b(ers)313 b(of)h(the)f(\014rst)g(kind)h (are)g(not)f(the)h(ones)g(app)26 b(earing)315 b(in)e([)p 0 .5 0 TeXcolorrgb 36271 71296 a SDict begin H.S end 36271 71296 a 0 .5 0 TeXcolorrgb Fs(5)p 0 .5 0 TeXcolorrgb 36742 70725 a SDict begin H.R end 36742 70725 a 36742 71296 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Riordan ICA) cvn H.B /ANN pdfmark end 36742 71296 a Black Fs(],)i(p.)e(75,)i(table)g (2.)p Black Black 26523 74617 a FA(5)p Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -2672 a SDict begin H.S end 800 -2672 a 0 TeXcolorgray 0 TeXcolorgray 800 -2672 a SDict begin H.R end 800 -2672 a 800 -2672 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 800 -2672 a Black Black 3875 x FA(F)-92 b(or)368 b FB(l)330 b Fz(2)307 b Fy(N)370 b FA(the)f(function)i FB(c)p FA(1\()p FB(l)22 b FA(;)184 b FB(y)40 b FA(\))371 b(generates)e(the)h(n)-31 b(um)g(b)31 b(ers)27030 -126 y SDict begin H.S end 27030 -126 a 27030 -126 a SDict begin 12 H.A end 27030 -126 a 27030 -126 a SDict begin [ /View [/XYZ H.V] /Dest (equation.43) cvn H.B /DEST pdfmark end 27030 -126 a 15008 4511 a FB(c)p FA(1\()p FB(l)22 b FA(;)184 b FB(y)40 b FA(\))617 b(=)20812 3127 y Fq(1)20454 3459 y Fv(X)20419 5803 y Fx(n)p Fu(=0)22456 4511 y FB(c)p FA(1)23488 4054 y Fu(\()p Fx(l)t Fu(\))23488 4784 y Fx(n)24705 4511 y FB(y)25288 4054 y Fx(n)26259 4511 y FB(;)923 b(c)p FA(1)28521 4054 y Fu(\()p Fx(l)t Fu(\))28521 4784 y Fx(n)30168 4511 y FA(=)31644 2950 y Fv(\022)32459 3762 y FB(n)246 b FA(+)g FB(l)32514 5270 y(l)268 b Fz(\000)246 b FA(1)34828 2950 y Fv(\023)35828 4511 y FB(l)36180 4054 y Fx(n)p Fq(\000)p Fu(1)38284 4511 y FB(:)12241 b FA(\(43\))800 7947 y(F)-92 b(or)361 b FB(l)330 b FA(=)307 b(2)363 b(this)f(is)g(the)g FC(EIS)g FA(sequence)p 0 1 0 0 TeXcolorcmyk 17422 8169 a SDict begin H.S end 17422 8169 a 0 1 0 0 TeXcolorcmyk -222 x FA(A001792)p 17422 8124 4152 45 v 0 1 0 0 TeXcolorcmyk 21573 7191 a SDict begin H.R end 21573 7191 a 21573 7947 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A001792) >> /Subtype /Link H.B /ANN pdfmark end 21573 7947 a Black 362 w Fz(f)p FA(1)p FB(;)184 b FA(3)p FB(;)g FA(8)p FB(;)g FA(20)p FB(;)g FA(48)p FB(;)g(:::)p Fz(g)p FA(.)500 b(F)-92 b(or)362 b(negativ)-31 b(e)364 b FB(l)22 b FA(,)364 b FB(c)p FA(1\()p FB(l)22 b FA(;)184 b FB(y)40 b FA(\))363 b(b)31 b(ecomes)362 b(a)h(p)31 b(olynomial)800 9276 y(in)371 b FB(y)40 b FA(;)371 b FC(e.g.)498 b FB(c)p FA(1\()p Fz(\000)p FA(2;)184 b FB(y)40 b FA(\))312 b(=)e(1)248 b(+)f FB(y)40 b FA(,)371 b(or)f FB(g)40 b FA(1\()p Fz(\000)p FA(1;)184 b FB(y)40 b FA(\))312 b(=)e FB(y)286 b FA(+)247 b FB(y)24410 8874 y Fu(2)24906 9276 y FB(:)371 b FA(The)g(co)31 b(e\016cien)-31 b(ts)372 b(of)g(these)e(p)31 b(olynomials)374 b(de\014ne)c(a)h(triangle)h(of)800 10604 y(n)-31 b(um)g(b)31 b(ers)369 b(found)h(under)e(the)h FC(EIS)g FA(n)-31 b(um)g(b)31 b(er)p 0 1 0 0 TeXcolorcmyk 19620 10825 a SDict begin H.S end 19620 10825 a 0 1 0 0 TeXcolorcmyk -221 x FA(A049323)p 19620 10781 4152 45 v 0 1 0 0 TeXcolorcmyk 23771 9848 a SDict begin H.R end 23771 9848 a 23771 10604 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049323) >> /Subtype /Link H.B /ANN pdfmark end 23771 10604 a Black FA(.)492 b(Their)370 b(explicit)h(form)f(is,)f(for)g FB(l)330 b Fz(2)307 b Fy(N)p FA(,)40601 9276 y SDict begin H.S end 40601 9276 a 40601 9276 a SDict begin 12 H.A end 40601 9276 a 40601 9276 a SDict begin [ /View [/XYZ H.V] /Dest (equation.44) cvn H.B /DEST pdfmark end 40601 9276 a 11570 13634 a FB(c)p FA(1)12602 13176 y Fu(\()p Fq(\000)p Fx(l)t Fu(\))12602 13907 y Fx(n)14941 13634 y FA(=)16417 12073 y Fv(\022)18341 12885 y FB(l)17232 14393 y(n)246 b FA(+)g(1)19803 12073 y Fv(\023)20802 13634 y FB(l)21154 13176 y Fx(n)p Fq(\000)p Fu(1)23628 13634 y FA(for)369 b FB(n)308 b FA(=)g(0)p FB(;)184 b FA(1)p FB(;)g(:::;)g(l)273 b Fz(\000)246 b FA(1)p FB(;)554 b FA(and)370 b(0)f(otherwise)q FB(:)8803 b FA(\(44\))800 16718 y(Eq.)p 0 .5 0 TeXcolorrgb 3334 16718 a SDict begin H.S end 3334 16718 a 0 .5 0 TeXcolorrgb FA(43)p 0 .5 0 TeXcolorrgb 4441 16005 a SDict begin H.R end 4441 16005 a 4441 16718 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.43) cvn H.B /ANN pdfmark end 4441 16718 a Black 376 w FA(no)-31 b(w)378 b(implies,)h(using)e(eqs.)p 0 .5 0 TeXcolorrgb 16817 16718 a SDict begin H.S end 16817 16718 a 0 .5 0 TeXcolorrgb FA(41)p 0 .5 0 TeXcolorrgb 17924 16005 a SDict begin H.R end 17924 16005 a 17924 16718 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.41) cvn H.B /ANN pdfmark end 17924 16718 a Black 376 w FA(and)p 0 .5 0 TeXcolorrgb 20836 16718 a SDict begin H.S end 20836 16718 a 0 .5 0 TeXcolorrgb FA(40)p 0 .5 0 TeXcolorrgb 21943 16005 a SDict begin H.R end 21943 16005 a 21943 16718 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.40) cvn H.B /ANN pdfmark end 21943 16718 a Black FA(,)h(that)g FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))379 b(is)e(indeed)f(an)h(in)-31 b(teger)377 b(for)f(ev)-31 b(ery)376 b FB(k)355 b Fz(2)319 b Fy(Z)251 b Fz(n)g(f)p FA(1)p Fz(g)p FA(.)515 b(An)800 18047 y(explicit)371 b(form)f(for)f(the)h(en)-31 b(tries)369 b(in)g(the)h(\014rst)e(column)j (is)24938 16718 y SDict begin H.S end 24938 16718 a 24938 16718 a SDict begin 12 H.A end 24938 16718 a 24938 16718 a SDict begin [ /View [/XYZ H.V] /Dest (equation.45) cvn H.B /DEST pdfmark end 24938 16718 a 9211 21893 a FB(s)p FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g FA(1\))619 b(=)16048 19624 y Fv(8)16048 20620 y(<)16048 22613 y(:)17585 19598 y(\000)18093 20003 y Fx(k)19 b Fq(\000)p Fu(2)p Fq(\000)p Fx(n)18712 20876 y(k)g Fq(\000)p Fu(2)20954 19598 y Fv(\001)21646 20494 y FA(\()p FB(k)282 b Fz(\000)246 b FA(1\))25024 20093 y Fx(n)p Fq(\000)p Fu(2)27867 20494 y FA(for)369 b FB(k)343 b FA(=)308 b(2)p FB(;)184 b FA(3)p FB(;)g(:::;)g FA(and)374 b FB(n)308 b Fz(2)g Fy(N)17585 22472 y Fv(\000)18093 22878 y Fq(j)p Fx(k)19 b Fq(j)p Fu(+1)18887 23751 y Fx(n)20229 22472 y Fv(\001)20920 23369 y FA(\()p Fz(j)p FB(k)35 b Fz(j)248 b FA(+)e(1\))24913 22967 y Fx(n)p Fq(\000)p Fu(2)27867 23369 y FA(for)369 b Fz(\000)p FB(k)343 b Fz(2)308 b Fy(N)33383 23535 y Fu(0)34064 23369 y FB(;)369 b(n)308 b FA(=)g(1)p FB(;)184 b FA(2)p FB(;)g(:::)p Fz(j)p FB(k)35 b Fz(j)251 b FA(+)246 b(1)370 b FB(:)50832 21893 y FA(\(45\))800 25739 y(The)323 b(e.g.f.s)h(for)e(the)h FB(m)p Fz(\000)p FA(th)g(column)g(of)g(the)g(signless)g FB(k)35 b Fz(\000)p FA(Stirling)324 b(n)-31 b(um)g(b)31 b(ers)322 b(of)h(the)g(\014rst)f(kind)h(are)f(then,)333 b(for)322 b FB(k)343 b FA(=)307 b(2)p FB(;)184 b FA(3)p FB(;)g(:::)p FA(,)14919 28986 y SDict begin H.S end 14919 28986 a 14919 28986 a SDict begin 12 H.A end 14919 28986 a 14919 28986 a SDict begin [ /View [/XYZ H.V] /Dest (equation.46) cvn H.B /DEST pdfmark end 14919 28986 a FB(G)p FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b FB(m)p FA(;)g FB(x)p FA(\))1418 b(:=)25522 27602 y Fq(1)25164 27934 y Fv(X)24957 30259 y Fx(n)p Fu(=)p Fx(m)27339 28986 y FB(S)64 b FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))33601 28237 y FB(x)34234 27835 y Fx(n)p 33601 28731 1236 45 v 33733 29745 a FB(n)p FA(!)35339 28986 y FB(;)p Black 15186 w FA(\(46\))p Black 22528 32274 a(=)25453 31526 y(1)p 25090 32020 1280 45 v 25090 33034 a FB(m)p FA(!)26687 31046 y Fv(h)27210 32274 y FA(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(g)40 b FA(1\()p FB(k)35 b FA(;)34503 31526 y FB(x)p 33560 32020 2518 45 v 33560 33034 a(k)282 b Fz(\000)246 b FA(1)36211 32274 y(\))36641 31046 y Fv(i)37164 31275 y Fx(m)38374 32274 y FB(:)p Black 12151 w FA(\(47\))p Black 800 35159 a(The)393 b(case)f FB(k)382 b FA(=)346 b(1)393 b(corresp)31 b(onds)391 b(to)j(the)f(ordinary)g(unsigned)g (Stirling)h(n)-31 b(um)g(b)31 b(ers)392 b FB(S)64 b FA(1)p FB(p)p FA(\()p FB(n;)184 b(m)p FA(\))395 b(with)e(e.g.f.)565 b(for)392 b(column)i FB(m)800 36488 y FA(giv)-31 b(en)370 b(b)-31 b(y)370 b FB(G)p FA(1)p FB(p)p FA(\(1;)184 b FB(m)p FA(;)g FB(x)p FA(\))619 b(=)13733 36052 y Fu(1)p 13435 36233 1037 45 v 13435 36869 a Fx(m)p Fu(!)14789 35591 y Fv(\000)15296 36488 y Fz(\000)p FB(l)22 b(n)p FA(\(1)248 b Fz(\000)e FB(x)p FA(\))20574 35591 y Fv(\001)21081 35820 y Fx(m)21922 36488 y FA(.)492 b(F)-92 b(rom)369 b(eqs.)p 0 .5 0 TeXcolorrgb 27820 36488 a SDict begin H.S end 27820 36488 a 0 .5 0 TeXcolorrgb FA(47)p 0 .5 0 TeXcolorrgb 28927 35774 a SDict begin H.R end 28927 35774 a 28927 36488 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.46) cvn H.B /ANN pdfmark end 28927 36488 a Black FA(,)p 0 .5 0 TeXcolorrgb 29972 36488 a SDict begin H.S end 29972 36488 a 0 .5 0 TeXcolorrgb FA(41)p 0 .5 0 TeXcolorrgb 31079 35774 a SDict begin H.R end 31079 35774 a 31079 36488 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.41) cvn H.B /ANN pdfmark end 31079 36488 a Black 369 w FA(and)p 0 .5 0 TeXcolorrgb 33969 36488 a SDict begin H.S end 33969 36488 a 0 .5 0 TeXcolorrgb FA(42)p 0 .5 0 TeXcolorrgb 35076 35774 a SDict begin H.R end 35076 35774 a 35076 36488 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.42) cvn H.B /ANN pdfmark end 35076 36488 a Black FA(,)35753 35159 y SDict begin H.S end 35753 35159 a 35753 35159 a SDict begin 12 H.A end 35753 35159 a 35753 35159 a SDict begin [ /View [/XYZ H.V] /Dest (equation.48) cvn H.B /DEST pdfmark end 35753 35159 a 10556 39540 a FB(G)p FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b(1;)g FB(x)p FA(\))619 b(=)c(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(g)40 b FA(1\()p FB(k)35 b FA(;)25573 38792 y FB(x)p 24620 39286 2540 45 v 24620 40300 a FA(1)247 b Fz(\000)f FB(x)27292 39540 y FA(\))616 b(=)30928 38792 y(1)p 29946 39286 2518 45 v 29946 40300 a FB(k)281 b Fz(\000)246 b FA(1)32781 39540 y(\()p Fz(\000)p FA(1)555 b(+)38989 38792 y(1)p 36727 39286 5078 45 v 36727 40300 a(\(1)247 b Fz(\000)f FB(x)p FA(\))40127 39980 y Fx(k)19 b Fq(\000)p Fu(1)41937 39540 y FA(\))370 b FB(;)7788 b FA(\(48\))800 42610 y(and)429 b(w)-31 b(e)429 b(reco)-31 b(v)g(er)429 b(the)g(result)g(for)g FB(k)442 b FA(=)406 b(1)429 b(from)h(l'H^)-553 b(opital's)432 b(rule)d(in)g(the)g(limit)i FB(k)321 b Fz(\000)285 b FA(1)408 b Fz(!)e FA(0.)672 b(Note)430 b(that)g FB(G)p FA(1\()p FB(k)35 b FA(;)184 b(1;)g FB(x)p FA(\))595 b(=)800 43938 y Fz(\000)p FB(G)p FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b(1;)g Fz(\000)p FB(x)p FA(\))496 b(=)p 9990 42975 5067 45 v 492 w FB(G)p FA(2\()p FB(k)35 b FA(;)184 b(1;)g FB(x)p FA(\))373 b(for)c FB(k)343 b Fz(2)307 b Fy(Z)p FA(.)800 45267 y(F)-92 b(or)357 b Fz(\000)p FB(k)343 b Fz(2)307 b Fy(N)6596 45433 y Fu(0)7450 45267 y FA(the)358 b(e.g.f.)491 b(for)358 b(the)g FB(m)p Fz(\000)p FA(th)g(column)h(of)f(the)g(nonnegativ)-31 b(e)360 b(triangular)g (matrix)f Fy(S1)q FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(\))359 b(is,)i(from)d(eqs.)p 0 .5 0 TeXcolorrgb 51693 45267 a SDict begin H.S end 51693 45267 a 0 .5 0 TeXcolorrgb FA(36)p 0 .5 0 TeXcolorrgb 52800 44553 a SDict begin H.R end 52800 44553 a 52800 45267 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.36) cvn H.B /ANN pdfmark end 52800 45267 a Black 800 46595 a FA(and)p 0 .5 0 TeXcolorrgb 3321 46595 a SDict begin H.S end 3321 46595 a 0 .5 0 TeXcolorrgb FA(46)p 0 .5 0 TeXcolorrgb 4428 45882 a SDict begin H.R end 4428 45882 a 4428 46595 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.46) cvn H.B /ANN pdfmark end 4428 46595 a Black FA(,)370 b FB(G)p FA(1\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(m)p FA(;)g FB(x)p FA(\))619 b(=)c(\()p Fz(\000)p FA(1\))16430 46193 y Fx(m)17457 46595 y FB(G)p FA(1)p FB(p)p FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(m)p FA(;)g Fz(\000)p FB(x)p FA(\),)374 b(hence)29587 45267 y SDict begin H.S end 29587 45267 a 29587 45267 a SDict begin 12 H.A end 29587 45267 a 29587 45267 a SDict begin [ /View [/XYZ H.V] /Dest (equation.49) cvn H.B /DEST pdfmark end 29587 45267 a 12185 49537 a FB(G)p FA(1\()p FB(k)35 b FA(;)184 b(1;)g FB(x)p FA(\))619 b(=)20764 48788 y(1)p 19474 49282 3133 45 v 19474 50296 a(1)247 b(+)f Fz(j)p FB(k)35 b Fz(j)22924 49537 y FA(\()p Fz(\000)p FA(1)555 b(+)e(\(1)247 b(+)f FB(x)p FA(\))30137 49080 y Fu(1+)p Fq(j)p Fx(k)19 b Fq(j)32328 49537 y FA(\))1108 b(for)984 b Fz(\000)246 b FB(k)343 b Fz(2)307 b Fy(N)40242 49703 y Fu(0)41108 49537 y FB(:)9417 b FA(\(49\))800 52657 y(F)-92 b(or)269 b(the)h(signed)g(matrix)h Fy(S1)q FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(\))272 b(with)f(elemen)-31 b(ts)271 b(de\014ned)e(b)-31 b(y)270 b FB(S)64 b FA(1)p FB(s)p FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(n;)g(m)p FA(\))312 b(:=)c(\()p Fz(\000)p FA(1\))39287 52256 y Fx(n)p Fq(\000)p Fx(m)41553 52657 y FB(S)64 b FA(1\()p Fz(\000j)p FB(k)35 b Fz(j)p FB(;)184 b(n;)g(m)p FA(\))274 b(the)c(e.g.f.)800 53986 y(of)406 b(the)f FB(m)p Fz(\000)p FA(th)g(column)i(is)e FB(G)p FA(1)p FB(s)p FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(m)p FA(;)g FB(x)p FA(\))739 b(=)c(\()p Fz(\000)p FA(1\))24466 53584 y Fx(m)25493 53986 y FB(G)p FA(1\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(m)p FA(;)g Fz(\000)p FB(x)p FA(\),)419 b FC(i.e.)601 b FB(G)p FA(1)p FB(s)p FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(x)p FA(\))555 b Fz(\021)d FB(G)p FA(1)p FB(s)p FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b(1;)g FB(x)p FA(\))740 b(=)800 55314 y(\(1)247 b Fz(\000)f FA(\(1)h Fz(\000)f FB(x)p FA(\))6537 54912 y Fu(1+)p Fq(j)p Fx(k)19 b Fq(j)8728 55314 y FA(\))p FB(=)p FA(\(1)248 b(+)e Fz(j)p FB(k)35 b Fz(j)p FA(\))371 b(for)e FB(k)343 b Fz(2)307 b Fy(N)18729 55480 y Fu(0)19226 55314 y FA(.)2460 56643 y(T)-92 b(ables)383 b(3)f(and)h(4)f(giv)-31 b(e)383 b(the)f FC(EIS)g FA(A-n)-31 b(um)g(b)31 b(ers)382 b(of)g(some)h(of)f(the)g(n)-31 b(um)g(b)31 b(er)382 b(triangles)i Fy(s1)p FA(\()p FB(k)35 b FA(\),)387 b Fy(S1)q FA(\()p FB(k)35 b FA(\))383 b(and)f Fy(S1p)q FA(\()p FB(k)35 b FA(\).)532 b(The)800 57971 y(A-n)-31 b(um)g(b)31 b(ers)369 b(of)h(the)f FB(m)308 b FA(=)f(1)370 b(column)g(and)f(of)h(the)f (sequence)g(of)h(ro)-31 b(w)370 b(sums)e(for)i(eac)-31 b(h)369 b(triangle)i(are)e(also)h(giv)-31 b(en)371 b(there.)2460 59299 y(The)i(o.g.f.)503 b(of)372 b(the)g(ro)-31 b(w)373 b(sequences)e(of)i(triangle)g Fy(S1)q FA(\()p FB(k)35 b FA(\))373 b(are)f(also)h(exp)31 b(onen)-31 b(tial)374 b(\(or)e(binomial\))j(con)-31 b(v)g(olution)376 b(p)31 b(olyno-)800 60628 y(mials.)494 b(In)369 b(other)g(w)-31 b(ords)370 b(the)f(p)31 b(olynomials)19643 59299 y SDict begin H.S end 19643 59299 a 19643 59299 a SDict begin 12 H.A end 19643 59299 a 19643 59299 a SDict begin [ /View [/XYZ H.V] /Dest (equation.50) cvn H.B /DEST pdfmark end 19643 59299 a 11836 63874 a FB(S)64 b FA(1)13132 64040 y Fx(n)13735 63874 y FA(\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))617 b(:=)19410 62490 y Fx(n)18884 62822 y Fv(X)18730 65167 y Fx(m)p Fu(=1)21005 63874 y FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))g FB(x)27207 63417 y Fx(m)28790 63874 y FB(;)923 b(S)64 b FA(1)31316 64040 y Fu(0)31813 63874 y FA(\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))309 b(:=)f(1)369 b FB(;)923 b(k)343 b Fz(2)307 b Fy(Z)369 b FB(;)9069 b FA(\(50\))800 67311 y(satisfy)484 b(eq.)p 0 .5 0 TeXcolorrgb 6236 67311 a SDict begin H.S end 6236 67311 a 0 .5 0 TeXcolorrgb FA(26)p 0 .5 0 TeXcolorrgb 7343 66597 a SDict begin H.R end 7343 66597 a 7343 67311 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.26) cvn H.B /ANN pdfmark end 7343 67311 a Black 482 w FA(with)g FB(S)64 b FA(2)483 b(replaced)f(b)-31 b(y)483 b FB(S)64 b FA(1.)833 b(In)482 b(the)h(notation)j(of)d(the)g(um)-31 b(bral)483 b(calculus)h(\()p FC(cf.)834 b FA([)p 0 .5 0 TeXcolorrgb 43219 67311 a SDict begin H.S end 43219 67311 a 0 .5 0 TeXcolorrgb FA(7)p 0 .5 0 TeXcolorrgb 43773 66597 a SDict begin H.R end 43773 66597 a 43773 67311 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Roman) cvn H.B /ANN pdfmark end 43773 67311 a Black FA(]\))483 b(the)g(p)31 b(olynomials)800 68639 y FB(S)64 b FA(1)2096 68805 y Fx(n)2699 68639 y FA(\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))452 b(are)f(a)g(sp)31 b(ecial)452 b(t)-31 b(yp)31 b(e)451 b(of)h(She\013er)d(p)31 b(olynomials)454 b(called)e(asso)31 b(ciated)452 b(p)31 b(olynomial)454 b(sequences)d(or)f(\\She\013er)h(for)800 69967 y(\(1)p FB(;)184 b(f)119 b FA(\()p FB(t)p FA(\)\).")1004 b(Here)538 b FB(f)119 b FA(\()p FB(t)p FA(\))1180 b(=)p 14417 69004 3788 45 v 1180 w FB(G)p FA(1\()p FB(k)35 b FA(;)184 b FB(t)p FA(\))r(\),)582 b(where)p 22893 69004 V 538 w FB(G)p FA(1\()p FB(k)35 b FA(;)184 b FB(t)p FA(\))1182 b(=)e FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(t)p FA(\))540 b(is)f(giv)-31 b(en,)582 b(for)539 b FB(k)625 b Fz(6)p FA(=)589 b(1,)582 b(in)539 b(eq.)p 0 .5 0 TeXcolorrgb 48258 69967 a SDict begin H.S end 48258 69967 a 0 .5 0 TeXcolorrgb FA(24)p 0 .5 0 TeXcolorrgb 49365 69254 a SDict begin H.R end 49365 69254 a 49365 69967 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.24) cvn H.B /ANN pdfmark end 49365 69967 a Black FA(.)1000 b(Also)p 800 70333 3730 45 v 800 71296 a FB(G)p FA(1\(1;)184 b FB(t)p FA(\))311 b(=)c FB(G)p FA(2\(1;)184 b FB(t)p FA(\))310 b(=)e FB(exp)p FA(\()p FB(t)p FA(\))247 b Fz(\000)f FA(1)369 b(is)g(obtained)i(in)e (the)h(limit)h(as)e(1)247 b Fz(\000)f FB(k)342 b Fz(!)308 b FA(0.)p Black 26523 74617 a(6)p Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -2672 a SDict begin H.S end 800 -2672 a 0 TeXcolorgray 0 TeXcolorgray 800 -2672 a SDict begin H.R end 800 -2672 a 800 -2672 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end 800 -2672 a Black Black 2460 1203 a FA(Eac)-31 b(h)411 b(sequence)e(of)h (ro)-31 b(w)410 b(sums)f(of)h(a)g(triangle)h(of)f(the)g(t)-31 b(yp)31 b(e)410 b(considered)f(in)h(this)g(w)-31 b(ork)410 b(is)g(generated)g(b)-31 b(y)410 b(a)f(function)800 2531 y(whic)-31 b(h)414 b(dep)31 b(ends)411 b(on)i(the)g(generating)i (function)f(of)f(the)g(triangle's)i(\014rst)d(\()p FB(m)380 b FA(=)g(1\))414 b(column.)624 b(F)-92 b(or)412 b(the)h Fy(s2)q FA(\()p FB(k)35 b FA(\))413 b(and)g Fy(s1)q FA(\()p FB(k)35 b FA(\))800 3859 y(triangles,)371 b(whic)-31 b(h)370 b(can)g(b)31 b(e)368 b(considered)h(as)h(Bell)g(matrices,)g (these)f(o.g.f.s)i(are,)f(for)f FB(k)343 b Fz(2)307 b Fy(Z)246 b Fz(n)h(f)p FA(1)p Fz(g)p FA(,)10120 7234 y SDict begin H.S end 10120 7234 a 10120 7234 a SDict begin 12 H.A end 10120 7234 a 10120 7234 a SDict begin [ /View [/XYZ H.V] /Dest (equation.51) cvn H.B /DEST pdfmark end 10120 7234 a FB(r)31 b FA(2\()p FB(k)k FA(;)184 b FB(x)p FA(\))1417 b(=)18576 6485 y FB(g)40 b FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))p 17623 6979 5625 45 v 17623 7993 a(1)247 b Fz(\000)f FB(g)40 b FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))23995 7234 y(=)25604 6485 y Fz(\000)p FA(1)554 b(+)f([1)247 b Fz(\000)f FA(\(1)h Fz(\000)f FB(k)35 b FA(\))34578 6084 y Fu(2)35260 6485 y FB(x)p FA(])36847 5710 y Fo(1)p 36333 5833 1405 38 v 36333 6274 a(1)p Fn(\000)p Fp(k)p 25604 6979 12322 45 v 26006 8245 a FB(k)588 b Fz(\000)554 b FA([1)247 b Fz(\000)f FA(\(1)h Fz(\000)f FB(k)35 b FA(\))34177 7925 y Fu(2)34858 8245 y FB(x)p FA(])36445 7469 y Fo(1)p 35931 7592 1405 38 v 35931 8033 a(1)p Fn(\000)p Fp(k)38059 7234 y FB(;)p Black 12466 w FA(\(51\))p Black 10120 11026 a FB(r)c FA(1\()p FB(k)k FA(;)184 b FB(x)p FA(\))1417 b(=)18576 10277 y FB(g)40 b FA(1\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))p 17623 10771 5625 45 v 17623 11785 a(1)247 b Fz(\000)f FB(g)40 b FA(1\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))23995 11026 y(=)29018 10277 y(1)553 b Fz(\000)h FA([1)247 b Fz(\000)f FA(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(x)p FA(])38254 9875 y Fx(k)19 b Fq(\000)p Fu(1)p 25604 10771 17744 45 v 25604 11785 a FA(\(1)247 b(+)f(\(1)h Fz(\000)f FB(k)35 b FA(\))31319 11466 y Fu(2)31816 11785 y FA(\))184 b([1)248 b Fz(\000)e FA(\()p FB(k)282 b Fz(\000)246 b FA(1\))184 b FB(x)p FA(])39147 11466 y Fx(k)19 b Fq(\000)p Fu(1)41379 11785 y Fz(\000)554 b FA(1)p Black 50832 11026 a(\(52\))p Black 800 14109 a(F)-92 b(or)371 b(the)h Fy(S2)q FA(\()p FB(k)35 b FA(\))373 b(\()p FB(k)347 b Fz(2)312 b Fy(N)11285 14275 y Fu(0)11782 14109 y FA(\))372 b(and)g Fy(S2p)q FA(\()p FB(k)35 b FA(\))373 b(\()p Fz(\000)p FB(k)347 b Fz(2)312 b Fy(N)22894 14275 y Fu(0)23391 14109 y FA(\))372 b(triangles,)j(whic)-31 b(h)373 b(can)f(b)31 b(e)371 b(in)-31 b(terpreted)372 b(as)g(Jab)31 b(otinsky)373 b(matrices,)800 15437 y(the)c(e.g.f.s)i(for)f(the)f(sequences)f(of)i (ro)-31 b(w)370 b(sums)f(are)10132 18002 y SDict begin H.S end 10132 18002 a 10132 18002 a SDict begin 12 H.A end 10132 18002 a 10132 18002 a SDict begin [ /View [/XYZ H.V] /Dest (equation.53) cvn H.B /DEST pdfmark end 10132 18002 a FB(R)9 b FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))1416 b(=)e FB(e)18336 17545 y Fx(G)p Fu(2\()p Fx(k)19 b Fu(;)p Fx(x)p Fu(\))22010 18002 y Fz(\000)553 b FA(1)616 b(=)f FB(exp)184 b FA([)p Fz(\000)p FA(1)247 b(+)f(\(1)h Fz(\000)f FA(\()p FB(k)281 b Fz(\000)246 b FA(1\))184 b FB(x)p FA(\))38643 17226 y Fo(1)p 38129 17349 1405 38 v 38129 17790 a(1)p Fn(\000)p Fp(k)39722 18002 y FA(])554 b Fz(\000)f FA(1)739 b FB(;)p Black 7236 w FA(\(53\))p Black 8099 21323 a FB(R)9 b FA(2)p FB(p)p FA(\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(x)p FA(\))1417 b(=)d FB(e)18336 20866 y Fx(G)p Fu(2)p Fx(p)p Fu(\()p Fq(\000j)p Fx(k)19 b Fq(j)p Fu(;)p Fx(x)p Fu(\))23684 21323 y Fz(\000)553 b FA(1)616 b(=)f FB(exp)184 b FA([1)247 b Fz(\000)f FA(\(1)h Fz(\000)f FA(\(1)h(+)f Fz(j)p FB(k)35 b Fz(j)p FA(\))184 b FB(x)p FA(\))40309 20490 y Fo(1)p 39557 20613 1882 38 v 39557 21065 a(1+)p Fn(j)p Fp(k)9 b Fn(j)41626 21323 y FA(])554 b Fz(\000)g FA(1)738 b FB(:)p Black 5332 w FA(\(54\))p Black 800 23757 a(F)-92 b(or)492 b(the)h Fy(S1p)q FA(\()p FB(k)35 b FA(\))493 b(\()p FB(k)549 b Fz(2)513 b Fy(N)12757 23923 y Fu(0)13254 23757 y FA(\))492 b(and)h Fy(S1)q FA(\()p FB(k)35 b FA(\))493 b(\()p Fz(\000)p FB(k)549 b Fz(2)513 b Fy(N)24423 23923 y Fu(0)24920 23757 y FA(\))493 b(triangles,)525 b(whic)-31 b(h)494 b(can)e(also)i(b)31 b(e)492 b(in)-31 b(terpreted)493 b(as)f(Jab)31 b(otinsky)800 25085 y(matrices,)371 b(the)e(e.g.f.s)i(for)e(the)h(sequences)e(of)i (ro)-31 b(w)369 b(sums)g(are)8921 27980 y SDict begin H.S end 8921 27980 a 8921 27980 a SDict begin 12 H.A end 8921 27980 a 8921 27980 a SDict begin [ /View [/XYZ H.V] /Dest (equation.55) cvn H.B /DEST pdfmark end 8921 27980 a FB(R)9 b FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))1416 b(=)e FB(e)17682 27523 y Fx(G)p Fu(1)p Fx(p)p Fu(\()p Fx(k)19 b Fu(;)p Fx(x)p Fu(\))21812 27980 y Fz(\000)553 b FA(1)616 b(=)f FB(exp)27576 26751 y Fv(\020)29352 27231 y FA(1)p 28370 27725 2518 45 v 28370 28739 a FB(k)281 b Fz(\000)246 b FA(1)31205 27980 y([)p Fz(\000)p FA(1)g(+)g(\(1)h Fz(\000)f FB(x)p FA(\))38326 27204 y Fo(1)p 37812 27327 1405 38 v 37812 27768 a Fp(k)9 b Fn(\000)p Fo(1)39405 27980 y FA(])39712 26751 y Fv(\021)40927 27980 y Fz(\000)554 b FA(1)738 b FB(;)p Black 6892 w FA(\(55\))p Black 8002 31545 a FB(R)9 b FA(1\()p Fz(\000j)p FB(k)35 b Fz(j)p FA(;)184 b FB(x)p FA(\))1417 b(=)d FB(e)17682 31088 y Fx(G)p Fu(1\()p Fq(\000j)p Fx(k)19 b Fq(j)p Fu(;)p Fx(x)p Fu(\))22573 31545 y Fz(\000)554 b FA(1)615 b(=)g FB(exp)28337 30316 y Fv(\020)30421 30796 y FA(1)p 29131 31291 3133 45 v 29131 32304 a(1)246 b(+)g Fz(j)p FB(k)35 b Fz(j)32396 31545 y FA([)p Fz(\000)p FA(1)247 b(+)f(\(1)h(+)f FB(x)p FA(\))38871 31088 y Fu(1+)p Fq(j)p Fx(k)19 b Fq(j)41063 31545 y FA(])41370 30316 y Fv(\021)42585 31545 y Fz(\000)553 b FA(1)739 b FB(:)p Black 5234 w FA(\(56\))p Black 800 34628 a(The)369 b(sp)31 b(ecial)370 b(case)f FB(k)343 b FA(=)307 b(1)370 b(can)f(b)31 b(e)369 b(obtained)i(for)e FB(R)9 b FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))370 b(and)g FB(R)9 b FA(1)p FB(p)p FA(\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))370 b(b)-31 b(y)369 b(taking)j(the)d(limit)i(as)f FB(k)281 b Fz(\000)246 b FA(1)308 b Fz(!)f FA(0.)2460 35957 y(In)369 b(Sections)h(2)g(and)f(3)h(w)-31 b(e)369 b(will)i(giv)-31 b(e)371 b(pro)31 b(ofs)369 b(of)g(the)h(results)f (stated)h(ab)31 b(o)-31 b(v)g(e.)800 37840 y SDict begin H.S end 800 37840 a 800 37840 a SDict begin 12 H.A end 800 37840 a 800 37840 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 800 37840 a 1777 x Fw(2)1793 b(k)p Fl(\000)p Fw(Stirling)599 b(n)-50 b(um)g(b)50 b(ers)600 b(of)e(the)g(second)h(kind)800 42042 y Fy(De\014nition)401 b(1:)482 b FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\).)489 b(The)349 b FB(k)35 b FA(-)p FC(Stirling)378 b(numb)-57 b(ers)377 b(of)g(the)g(se)-57 b(c)g(ond)377 b(kind)p FA(,)354 b FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\),)356 b(are)348 b(de\014ned)g(for)h FB(k)342 b Fz(2)308 b Fy(Z)348 b FA(b)-31 b(y)800 43370 y(eq.)p 0 .5 0 TeXcolorrgb 2553 43370 a SDict begin H.S end 2553 43370 a 0 .5 0 TeXcolorrgb FA(12)p 0 .5 0 TeXcolorrgb 3660 42657 a SDict begin H.R end 3660 42657 a 3660 43370 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.12) cvn H.B /ANN pdfmark end 3660 43370 a Black FA(.)800 45498 y Fy(Lemma)424 b(1:)492 b FA(The)369 b(n)-31 b(um)g(b)31 b(ers)369 b FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(satisfy)d(the)f(recursion)g(relation)i(eq.)p 0 .5 0 TeXcolorrgb 35222 45498 a SDict begin H.S end 35222 45498 a 0 .5 0 TeXcolorrgb FA(13)p 0 .5 0 TeXcolorrgb 36329 44785 a SDict begin H.R end 36329 44785 a 36329 45498 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 36329 45498 a Black FA(.)800 47627 y FC(Pr)-57 b(o)g(of)p FA(:)768 b(Consider)508 b(\()p FB(x)10273 47225 y Fx(k)11002 47627 y FB(d)11578 47793 y Fx(x)12135 47627 y FA(\))12565 47225 y Fx(n)14242 47627 y FA(=)1074 b FB(x)16810 47225 y Fx(k)17539 47627 y FB(d)18115 47793 y Fx(x)18857 47627 y FA(\()p FB(x)19920 47225 y Fx(k)20649 47627 y FB(d)21225 47793 y Fx(x)21783 47627 y FA(\))22213 47225 y Fx(n)p Fq(\000)p Fu(1)24455 47627 y FA(and)508 b(use)e(eq.)p 0 .5 0 TeXcolorrgb 30686 47627 a SDict begin H.S end 30686 47627 a 0 .5 0 TeXcolorrgb FA(12)p 0 .5 0 TeXcolorrgb 31793 46913 a SDict begin H.R end 31793 46913 a 31793 47627 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.12) cvn H.B /ANN pdfmark end 31793 47627 a Black 507 w FA(with)i FB(n)537 b Fz(!)g FB(n)338 b Fz(\000)g FA(1)507 b(together)h(with)g(the)g(lo)-31 b(w)g(er)800 48955 y(triangular)371 b(matrix)g(conditions)g(giv)-31 b(en)370 b(after)g(this)f(eq.)493 b(Then)369 b(compare)h(co)31 b(e\016cien)-31 b(ts)371 b(of)f Fz(f)p FB(x)39780 48553 y Fx(m)40805 48955 y FB(d)41750 48553 y Fx(m)41381 49229 y(x)42591 48955 y Fz(g)43144 48553 y Fx(n)43144 49230 y Fu(1)43747 48955 y FA(.)p 44793 48402 687 67 v 44793 48955 67 554 v 45412 48955 V 44793 49021 687 67 v 800 51083 a Fy(Note)426 b(1:)492 b FA(It)369 b(follo)-31 b(ws)371 b(from)f(eq.)p 0 .5 0 TeXcolorrgb 14621 51083 a SDict begin H.S end 14621 51083 a 0 .5 0 TeXcolorrgb FA(13)p 0 .5 0 TeXcolorrgb 15728 50370 a SDict begin H.R end 15728 50370 a 15728 51083 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 15728 51083 a Black 369 w FA(and)f(the)h(initial)i(conditions)f(that)f(the)f FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(are)c(alw)-31 b(a)g(ys)371 b(in)-31 b(tegers.)800 53212 y Fy(De\014nition)394 b(2:)479 b FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\).)487 b(The)343 b FC(asso)-57 b(ciate)g(d)341 b FB(k)35 b FA(-)p FC(Stirling)373 b(numb)-57 b(ers)372 b(of)f(the)g(se)-57 b(c)g(ond)372 b(kind)p FA(,)348 b FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\),)352 b(are)342 b(de\014ned)g(for)800 54540 y FB(k)h Fz(2)307 b Fy(Z)246 b Fz(n)h(f)p FA(1)p Fz(g)370 b FA(b)-31 b(y)369 b(eq.)p 0 .5 0 TeXcolorrgb 9907 54540 a SDict begin H.S end 9907 54540 a 0 .5 0 TeXcolorrgb FA(18)p 0 .5 0 TeXcolorrgb 11014 53827 a SDict begin H.R end 11014 53827 a 11014 54540 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.18) cvn H.B /ANN pdfmark end 11014 54540 a Black FA(.)800 56669 y Fy(Lemma)424 b(2:)492 b FA(The)369 b(n)-31 b(um)g(b)31 b(ers)369 b FB(s)p FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))373 b(satisfy)d(the)g(recursion)e (relation)j(giv)-31 b(en)371 b(b)-31 b(y)369 b(eqs.)p 0 .5 0 TeXcolorrgb 39862 56669 a SDict begin H.S end 39862 56669 a 0 .5 0 TeXcolorrgb FA(19)p 0 .5 0 TeXcolorrgb 40969 55955 a SDict begin H.R end 40969 55955 a 40969 56669 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.19) cvn 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57340 y Fx(n)40373 57174 y FA(\()p FB(x)p FA(\))265 b(as)g(ro)-31 b(w)266 b(\(or)g(Jab)31 b(otinsky\))800 58503 y(p)g(olynomials.)495 b(Kn)-31 b(uth's)370 b FB(f)119 b FA(\()p FB(z)49 b FA(\))370 b(corresp)31 b(onds)368 b(to)i(our)f(e.g.f.)494 b(for)369 b(the)h FB(m)307 b FA(=)g(1)370 b(column)g(sequence.\))800 60631 y(Another)389 b(notation)i(is)e(used)f(in)g(the)h(um)-31 b(bral)390 b(calculus)f(\()p FC(cf.)551 b FA([)p 0 .5 0 TeXcolorrgb 27008 60631 a SDict begin H.S end 27008 60631 a 0 .5 0 TeXcolorrgb FA(7)p 0 .5 0 TeXcolorrgb 27562 59918 a SDict begin H.R end 27562 59918 a 27562 60631 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Roman) cvn H.B /ANN pdfmark end 27562 60631 a Black FA(]\).)h(The)388 b(ro)-31 b(w)390 b(p)31 b(olynomials)391 b FB(E)40589 60797 y Fx(n)41192 60631 y FA(\()p FB(x)p FA(\))524 b(=)44594 59801 y Fv(P)45763 60074 y Fx(m)45763 60963 y(n)p Fu(=1)47671 60631 y FB(J)106 b FA(\()p FB(n;)184 b(m)p FA(\))g FB(x)52195 60229 y 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FA(\()p FB(m)p FA(;)184 b FB(x)p FA(\).)p 17092 33334 687 67 v 17092 33888 67 554 v 17712 33888 V 17092 33954 687 67 v 800 36016 a Fy(Lemma)381 b(12:)474 b FA(Ro)-31 b(w)332 b(sums)f(of)h(exp)31 b(onen)-31 b(tial)334 b(con)-31 b(v)g(olution)335 b(matrices.)481 b(The)332 b(e.g.f.)482 b FB(R)9 b FA(\()p FB(x)p FA(\))491 b(:=)39972 35186 y Fv(P)41140 35459 y Fq(1)41140 36348 y Fx(n)p Fu(=1)43233 36016 y FB(R)44074 36182 y Fx(n)44860 36016 y FB(x)45493 35614 y Fx(n)46095 36016 y FB(=n)p FA(!)333 b(of)f(the)g(ro)-31 b(w)800 37344 y(sums)386 b FB(R)4438 37510 y Fx(n)5561 37344 y FA(:=)7251 36514 y Fv(P)8420 36788 y Fx(n)8420 37676 y(m)p Fu(=1)10750 37344 y FB(S)64 b FA(\()p FB(n;)184 b(m)p FA(\))389 b(of)e(a)g(lo)-31 b(w)g(er)388 b(triangular)g(exp)31 b(onen)-31 b(tial)389 b(con)-31 b(v)g(olution)390 b(matrix)f Fz(f)p FB(S)64 b FA(\()p FB(n;)184 b(m)p FA(\))p Fz(g)45546 37510 y Fx(n)p Fq(\025)p Fx(m)p Fq(\025)p Fu(1)49148 37344 y FA(is)387 b(giv)-31 b(en)800 38673 y(b)g(y)2337 37344 y SDict begin H.S end 2337 37344 a 2337 37344 a SDict begin 12 H.A end 2337 37344 a 2337 37344 a SDict begin [ /View [/XYZ H.V] /Dest (equation.67) cvn H.B /DEST pdfmark end 2337 37344 a 21604 40001 a FB(R)9 b FA(\()p FB(x)p FA(\))614 b(:=)i FB(e)26860 39544 y Fx(G)p Fu(\()p Fx(x)p Fu(\))29352 40001 y Fz(\000)553 b FA(1)370 b FB(;)18836 b FA(\(67\))800 41869 y(where)369 b FB(G)p FA(\()p FB(x)p FA(\))h(is)f(the)h(e.g.f.)494 b(of)369 b(the)h(\014rst)e(\()p FB(m)308 b FA(=)g(1\))370 b(column)g(sequence)f(of)h(the)f(matrix)i Fz(f)p FB(S)64 b FA(\()p FB(n;)184 b(m)p FA(\))p Fz(g)42301 42035 y Fx(n)p Fq(\025)p Fx(m)p Fq(\025)p Fu(1)45517 41869 y FA(.)800 43998 y FC(Pr)-57 b(o)g(of)p FA(:)493 b(Analogous)371 b(to)f(the)g(pro)31 b(of)369 b(of)h(Lemma)g(11.)p 23079 43444 687 67 v 23079 43998 67 554 v 23699 43998 V 23079 44064 687 67 v 800 46126 a Fy(Prop)35 b(osition)395 b(2:)478 b FA(O.g.f.)485 b(for)341 b(ro)-31 b(w)342 b(sums)f(of)h(the)f Fy(s2)q FA(\()p FB(k)35 b FA(\))342 b(triangles.)485 b(F)-92 b(or)340 b FB(k)j Fz(2)307 b Fy(Z)191 b Fz(n)g(f)p FA(1)p Fz(g)342 b FA(the)g(o.g.f.)485 b(of)342 b(the)g(sequence)e(of)i (ro)-31 b(w)800 47454 y(sums)369 b(of)g(the)h(lo)-31 b(w)g(er)370 b(triangular)h(matrix)g Fy(s2)q FA(\()p FB(k)35 b FA(\))370 b(is)f(giv)-31 b(en)370 b(b)-31 b(y)370 b(eq.)p 0 .5 0 TeXcolorrgb 28686 47454 a SDict begin H.S end 28686 47454 a 0 .5 0 TeXcolorrgb FA(51)p 0 .5 0 TeXcolorrgb 29793 46741 a SDict begin H.R end 29793 46741 a 29793 47454 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.51) cvn H.B /ANN pdfmark end 29793 47454 a Black FA(.)800 49583 y FC(Pr)-57 b(o)g(of)p FA(:)493 b(Lemma)370 b(11)g(and)g(the)f FB(g)40 b FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\))371 b(result)e(from)h(Lemma)g(6,)g(eq.)p 0 .5 0 TeXcolorrgb 30383 49583 a SDict begin H.S end 30383 49583 a 0 .5 0 TeXcolorrgb FA(60)p 0 .5 0 TeXcolorrgb 31490 48869 a SDict begin H.R end 31490 48869 a 31490 49583 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.60) cvn H.B /ANN pdfmark end 31490 49583 a Black FA(.)p 32535 49029 687 67 v 32535 49583 67 554 v 33155 49583 V 32535 49649 687 67 v 800 51711 a Fy(Prop)35 b(osition)494 b(3:)608 b FA(E.g.f.)670 b(for)428 b(the)f(sequence)g(of)h(ro)-31 b(w)429 b(sums)e(of)h(the)f Fy(S2)q FA(\()p FB(k)35 b FA(\))429 b(and)e Fy(S2p)q FA(\()p FB(k)35 b FA(\))429 b(triangles.)669 b(F)-92 b(or)427 b FB(k)439 b Fz(2)405 b Fy(N)50278 51877 y Fk(0)51263 51711 y FA(the)800 53039 y(e.g.f.)551 b(of)388 b(the)h(sequence)e(of)i(ro)-31 b(w)388 b(sums)g(of)g(the)h(nonnegativ)-31 b(e)390 b(lo)-31 b(w)g(er)390 b(triangular)f(matrix)h Fy(S2)q FA(\()p FB(k)35 b FA(\),)394 b(resp.)548 b Fy(S2p)q FA(\()p FB(k)35 b FA(\),)394 b(de\014ned)800 54368 y(from)370 b(eq.)p 0 .5 0 TeXcolorrgb 5169 54368 a SDict begin H.S end 5169 54368 a 0 .5 0 TeXcolorrgb FA(13)p 0 .5 0 TeXcolorrgb 6276 53654 a SDict begin H.R end 6276 53654 a 6276 54368 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 6276 54368 a Black FA(,)g(resp.)492 b(eq.)p 0 .5 0 TeXcolorrgb 11482 54368 a SDict begin H.S end 11482 54368 a 0 .5 0 TeXcolorrgb FA(27)p 0 .5 0 TeXcolorrgb 12589 53654 a SDict begin H.R end 12589 53654 a 12589 54368 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.27) cvn H.B /ANN pdfmark end 12589 54368 a Black FA(,)370 b(is)f(giv)-31 b(en)370 b(b)-31 b(y)370 b(eq.)p 0 .5 0 TeXcolorrgb 20559 54368 a SDict begin H.S end 20559 54368 a 0 .5 0 TeXcolorrgb FA(53)p 0 .5 0 TeXcolorrgb 21666 53654 a SDict begin H.R end 21666 53654 a 21666 54368 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.53) cvn H.B /ANN pdfmark end 21666 54368 a Black FA(,)g(resp.)491 b(eq.)p 0 .5 0 TeXcolorrgb 26872 54368 a SDict begin H.S end 26872 54368 a 0 .5 0 TeXcolorrgb FA(54)p 0 .5 0 TeXcolorrgb 27979 53654 a SDict begin H.R end 27979 53654 a 27979 54368 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.53) cvn H.B /ANN pdfmark end 27979 54368 a Black FA(.)800 56496 y FC(Pr)-57 b(o)g(of)p FA(:)493 b(Lemma)370 b(12)g(and)g FB(G)p FA(2\()p FB(k)35 b FA(;)184 b FB(x)p FA(\),)372 b(resp.)491 b FB(G)p FA(2)p FB(p)p FA(\()p Fz(\000)p FB(k)35 b FA(;)184 b FB(x)p FA(\),)372 b(from)e(eq.)p 0 .5 0 TeXcolorrgb 29713 56496 a SDict begin H.S end 29713 56496 a 0 .5 0 TeXcolorrgb FA(24)p 0 .5 0 TeXcolorrgb 30820 55783 a SDict begin H.R end 30820 55783 a 30820 56496 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.24) cvn H.B /ANN pdfmark end 30820 56496 a Black FA(,)f(resp.)492 b(eq.)p 0 .5 0 TeXcolorrgb 36025 56496 a SDict begin H.S end 36025 56496 a 0 .5 0 TeXcolorrgb FA(29)p 0 .5 0 TeXcolorrgb 37132 55783 a SDict begin H.R end 37132 55783 a 37132 56496 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.29) cvn H.B /ANN pdfmark end 37132 56496 a Black FA(.)p 38178 55943 687 67 v 38178 56496 67 554 v 38798 56496 V 38178 56562 687 67 v 800 58405 a SDict begin H.S end 800 58405 a 800 58405 a SDict begin 12 H.A end 800 58405 a 800 58405 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 800 58405 a 1716 x Fw(3)1793 b(k)p Fl(\000)p Fw(Stirling)599 b(n)-50 b(um)g(b)50 b(ers)600 b(of)e(the)g(\014rst)i(kind)4271 62546 y FB(k)35 b Fz(\000)p FA(Stirling)323 b(n)-31 b(um)g(b)31 b(ers)321 b(of)h(the)f(\014rst)g(kind)h(can)g(b)31 b(e)320 b(de\014ned)h(as)h (the)f(elemen)-31 b(ts)323 b(of)e(the)h(\(in\014nite-dimensional,)336 b(lo)-31 b(w)g(er)800 63874 y(triangular\))465 b(in)-31 b(v)g(erse)463 b(matrix)h Fy(S1)q FA(\()p FB(k)35 b FA(\))463 b(to)h(the)e(matrix)j Fy(S2)p FA(\()p FB(k)35 b FA(\))464 b(formed)f(from)g(the)g FB(k)35 b Fz(\000)p FA(Stirling)465 b(n)-31 b(um)g(b)31 b(ers)462 b(of)h(the)g(second)800 65202 y(kind.)800 67331 y Fy(De\014nition)426 b(3:)492 b FB(k)35 b Fz(\000)p FC(Stirling)397 b(numb)-57 b(ers)397 b(of)f(the)g(\014rst)g(kind)p 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Fz(O)13711 20966 y Fx(x)14268 20800 y FA(\()p FB(k)35 b FA(;)184 b FB(n)p FA(\))618 b(:=)d FB(x)19928 20343 y Fq(\000)p Fu(\()p Fx(k)19 b Fq(\000)p Fu(1\))p Fx(n)23721 20800 y FA(\()p FB(x)24784 20343 y Fx(k)25513 20800 y FB(d)26089 20966 y Fx(x)26647 20800 y FA(\))27077 20343 y Fx(n)28295 20800 y FA(=)30297 19417 y Fx(n)29771 19749 y Fv(X)29782 22093 y Fx(p)p Fu(=1)31739 20800 y FB(\016)42 b FA(\()p FB(k)35 b FA(;)184 b FB(n;)g(p)p FA(\))g Fz(O)37012 20966 y Fx(x)37573 20800 y FA(\()p FB(k)35 b FA(;)184 b FB(p)p FA(\))371 b FB(;)10062 b FA(\(70\))800 24437 y(with)542 b FB(\016)42 b FA(\()p FB(k)35 b FA(;)184 b FB(n;)g(p)p FA(\))597 b(:=)10060 23606 y Fv(P)11228 23880 y Fx(n)11228 24769 y(m)p Fu(=)p Fx(p)13574 24437 y FB(S)64 b FA(2\()p FB(k)35 b FA(;)184 b FB(n;)g(m)p FA(\))g FB(S)64 b FA(1\()p FB(k)35 b FA(;)184 b FB(m;)g(p)p FA(\).)1014 b(Since)541 b(the)h(op)31 b(erators)541 b Fz(fO)37390 24603 y Fx(x)37948 24437 y FA(\()p FB(k)35 b FA(;)184 b FB(p)p FA(\))p Fz(g)41020 24035 y 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28859 30652 a SDict begin H.R end 28859 30652 a 28859 31365 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.68) cvn H.B /ANN pdfmark end 28859 31365 a Black FA(.)912 b(No)-31 b(w)510 b(w)-31 b(e)509 b(compare)h(co)31 b(e\016cien)-31 b(ts)510 b(of)g(the)f(op)31 b(erators)800 32693 y Fz(f)p FB(x)1986 32292 y Fx(p)2682 32693 y FB(d)3627 32292 y Fx(p)3258 32967 y(x)4139 32693 y Fz(g)4692 32292 y Fx(n)4692 32968 y Fu(1)5295 32693 y FA(.)800 34822 y Fy(Lemma)424 b(14:)493 b FA(The)369 b FB(k)35 b Fz(\000)p FA(Stirling)371 b(n)-31 b(um)g(b)31 b(ers)369 b(of)h(the)f(\014rst)g(kind)g(satisfy)h (the)g(recurrence)d(giv)-31 b(en)371 b(in)e(eq.)p 0 .5 0 TeXcolorrgb 44186 34822 a SDict begin H.S end 44186 34822 a 0 .5 0 TeXcolorrgb FA(37)p 0 .5 0 TeXcolorrgb 45293 34108 a SDict begin H.R end 45293 34108 a 45293 34822 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.37) cvn H.B /ANN pdfmark end 45293 34822 a Black FA(.)800 36950 y FC(Pr)-57 b(o)g(of)p FA(:)703 b(Use)473 b FB(x)7250 36548 y Fx(n)8037 36950 y FB(d)8982 36548 y Fx(n)8613 37224 y(x)10549 36950 y FA(=)965 b(\()p FB(x)184 b(d)14198 37116 y Fx(x)15072 36950 y Fz(\000)316 b FA(\()p FB(n)h Fz(\000)f FA(1\)\))184 b FB(x)21067 36548 y Fx(n)p Fq(\000)p Fu(1)22988 36950 y FB(d)23933 36548 y Fx(n)p Fq(\000)p Fu(1)23564 37224 y Fx(x)26143 36950 y FA(and)474 b(insert)g(eq.)p 0 .5 0 TeXcolorrgb 33447 36950 a SDict begin H.S end 33447 36950 a 0 .5 0 TeXcolorrgb FA(68)p 0 .5 0 TeXcolorrgb 34554 36237 a SDict begin H.R end 34554 36237 a 34554 36950 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.68) cvn H.B /ANN pdfmark end 34554 36950 a Black 474 w FA(in)g(b)31 b(oth)474 b(sides)g(of)h(this)f(iden)-31 b(tit)g(y)-92 b(.)810 b(After)800 38279 y(di\013eren)-31 b(tiation,)380 b(remem)-31 b(b)31 b(ering)377 b(the)f(triangularit)-31 b(y)379 b(of)d Fy(S1)q FA(\()p FB(k)35 b FA(\),)378 b(w)-31 b(e)377 b(compare)f(co)31 b(e\016cien)-31 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(http://oeis.org/A011801) >> /Subtype /Link H.B /ANN pdfmark end 52757 10178 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 800 11717 a SDict begin H.S end 800 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A013988)p 800 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 4640 10826 a SDict begin H.R end 4640 10826 a 4640 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A013988) >> /Subtype /Link H.B /ANN pdfmark end 4640 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5174 11717 a SDict begin H.S end 5174 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A015735)p 5174 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 9014 10826 a SDict begin H.R end 9014 10826 a 9014 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A015735) >> /Subtype /Link H.B /ANN pdfmark end 9014 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 9549 11717 a SDict begin H.S end 9549 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A016036)p 9549 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 13388 10826 a SDict begin H.R end 13388 10826 a 13388 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A016036) >> /Subtype /Link H.B /ANN pdfmark end 13388 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 13923 11717 a SDict begin H.S end 13923 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A019590)p 13923 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 17763 10826 a SDict begin H.R end 17763 10826 a 17763 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A019590) >> /Subtype /Link H.B /ANN pdfmark end 17763 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 18297 11717 a SDict begin H.S end 18297 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A023531)p 18297 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 22137 10826 a SDict begin H.R end 22137 10826 a 22137 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A023531) >> /Subtype /Link H.B /ANN pdfmark end 22137 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 22672 11717 a SDict begin H.S end 22672 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025748)p 22672 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 26511 10826 a SDict begin H.R end 26511 10826 a 26511 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025748) >> /Subtype /Link H.B /ANN pdfmark end 26511 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 27046 11717 a SDict begin H.S end 27046 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025748)p 27046 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 30885 10826 a SDict begin H.R end 30885 10826 a 30885 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025748) >> /Subtype /Link H.B /ANN pdfmark end 30885 11507 a Black Fa(-)p 0 1 0 0 TeXcolorcmyk 31226 11717 a SDict begin H.S end 31226 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025755)p 31226 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 35066 10826 a SDict begin H.R end 35066 10826 a 35066 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025755) >> /Subtype /Link H.B /ANN pdfmark end 35066 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 35601 11717 a SDict begin H.S end 35601 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025749)p 35601 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 39441 10826 a SDict begin H.R end 39441 10826 a 39441 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025749) >> /Subtype /Link H.B /ANN pdfmark end 39441 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 39975 11717 a SDict begin H.S end 39975 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025750)p 39975 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 43815 10826 a SDict begin H.R end 43815 10826 a 43815 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025750) >> /Subtype /Link H.B /ANN pdfmark end 43815 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 44350 11717 a SDict begin H.S end 44350 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025751)p 44350 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 48189 10826 a SDict begin H.R end 48189 10826 a 48189 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025751) >> /Subtype /Link H.B /ANN pdfmark end 48189 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 48724 11717 a SDict begin H.S end 48724 11717 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025756)p 48724 11675 3840 43 v 0 1 0 0 TeXcolorcmyk 52564 10826 a SDict begin H.R end 52564 10826 a 52564 11507 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025756) >> /Subtype /Link H.B /ANN pdfmark end 52564 11507 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 800 13045 a SDict begin H.S end 800 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025757)p 800 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 4640 12154 a SDict begin H.R end 4640 12154 a 4640 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025757) >> /Subtype /Link H.B /ANN pdfmark end 4640 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5174 13045 a SDict begin H.S end 5174 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025758)p 5174 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 9014 12154 a SDict begin H.R end 9014 12154 a 9014 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025758) >> /Subtype /Link H.B /ANN pdfmark end 9014 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 9549 13045 a SDict begin H.S end 9549 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A025759)p 9549 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 13388 12154 a SDict begin H.R end 13388 12154 a 13388 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A025759) >> /Subtype /Link H.B /ANN pdfmark end 13388 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 13923 13045 a SDict begin H.S end 13923 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A028575)p 13923 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 17763 12154 a SDict begin H.R end 17763 12154 a 17763 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A028575) >> /Subtype /Link H.B /ANN pdfmark end 17763 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 18297 13045 a SDict begin H.S end 18297 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A028844)p 18297 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 22137 12154 a SDict begin H.R end 22137 12154 a 22137 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A028844) >> /Subtype /Link H.B /ANN pdfmark end 22137 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 22672 13045 a SDict begin H.S end 22672 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A030523)p 22672 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 26511 12154 a SDict begin H.R end 26511 12154 a 26511 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A030523) >> /Subtype /Link H.B /ANN pdfmark end 26511 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 27046 13045 a SDict begin H.S end 27046 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A030524)p 27046 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 30885 12154 a SDict begin H.R end 30885 12154 a 30885 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A030524) >> /Subtype /Link H.B /ANN pdfmark end 30885 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 31420 13045 a SDict begin H.S end 31420 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A030526)p 31420 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 35260 12154 a SDict begin H.R end 35260 12154 a 35260 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A030526) >> /Subtype /Link H.B /ANN pdfmark end 35260 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 35795 13045 a SDict begin H.S end 35795 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A030527)p 35795 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 39634 12154 a SDict begin H.R end 39634 12154 a 39634 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A030527) >> /Subtype /Link H.B /ANN pdfmark end 39634 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 40169 13045 a SDict begin H.S end 40169 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A030528)p 40169 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 44008 12154 a SDict begin H.R end 44008 12154 a 44008 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A030528) >> /Subtype /Link H.B /ANN pdfmark end 44008 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 44543 13045 a SDict begin H.S end 44543 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A033184)p 44543 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 48383 12154 a SDict begin H.R end 48383 12154 a 48383 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A033184) >> /Subtype /Link H.B /ANN pdfmark end 48383 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 48917 13045 a SDict begin H.S end 48917 13045 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A033842)p 48917 13003 3840 43 v 0 1 0 0 TeXcolorcmyk 52757 12154 a SDict begin H.R end 52757 12154 a 52757 12835 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A033842) >> /Subtype /Link H.B /ANN pdfmark end 52757 12835 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 800 14374 a SDict begin H.S end 800 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A034171)p 800 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 4640 13483 a SDict begin H.R end 4640 13483 a 4640 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A034171) >> /Subtype /Link H.B /ANN pdfmark end 4640 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5174 14374 a SDict begin H.S end 5174 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A034255)p 5174 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 9014 13483 a SDict begin H.R end 9014 13483 a 9014 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A034255) >> /Subtype /Link H.B /ANN pdfmark end 9014 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 9549 14374 a SDict begin H.S end 9549 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A034687)p 9549 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 13388 13483 a SDict begin H.R end 13388 13483 a 13388 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A034687) >> /Subtype /Link H.B /ANN pdfmark end 13388 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 13923 14374 a SDict begin H.S end 13923 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A035323)p 13923 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 17763 13483 a SDict begin H.R end 17763 13483 a 17763 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A035323) >> /Subtype /Link H.B /ANN pdfmark end 17763 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 18297 14374 a SDict begin H.S end 18297 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A035324)p 18297 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 22137 13483 a SDict begin H.R end 22137 13483 a 22137 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A035324) >> /Subtype /Link H.B /ANN pdfmark end 22137 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 22672 14374 a SDict begin H.S end 22672 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A035342)p 22672 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 26511 13483 a SDict begin H.R end 26511 13483 a 26511 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A035342) >> /Subtype /Link H.B /ANN pdfmark end 26511 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 27046 14374 a SDict begin H.S end 27046 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A035469)p 27046 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 30885 13483 a SDict begin H.R end 30885 13483 a 30885 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A035469) >> /Subtype /Link H.B /ANN pdfmark end 30885 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 31420 14374 a SDict begin H.S end 31420 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A035529)p 31420 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 35260 13483 a SDict begin H.R end 35260 13483 a 35260 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A035529) >> /Subtype /Link H.B /ANN pdfmark end 35260 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 35795 14374 a SDict begin H.S end 35795 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A036068)p 35795 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 39634 13483 a SDict begin H.R end 39634 13483 a 39634 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A036068) >> /Subtype /Link H.B /ANN pdfmark end 39634 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 40169 14374 a SDict begin H.S end 40169 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A036070)p 40169 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 44008 13483 a SDict begin H.R end 44008 13483 a 44008 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A036070) >> /Subtype /Link H.B /ANN pdfmark end 44008 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 44543 14374 a SDict begin H.S end 44543 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A036083)p 44543 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 48383 13483 a SDict begin H.R end 48383 13483 a 48383 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A036083) >> /Subtype /Link H.B /ANN pdfmark end 48383 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 48917 14374 a SDict begin H.S end 48917 14374 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A039717)p 48917 14332 3840 43 v 0 1 0 0 TeXcolorcmyk 52757 13483 a SDict begin H.R end 52757 13483 a 52757 14164 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A039717) >> /Subtype /Link H.B /ANN pdfmark end 52757 14164 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 800 15702 a SDict begin H.S end 800 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A039746)p 800 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 4640 14811 a SDict begin H.R end 4640 14811 a 4640 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A039746) >> /Subtype /Link H.B /ANN pdfmark end 4640 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5174 15702 a SDict begin H.S end 5174 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A043553)p 5174 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 9014 14811 a SDict begin H.R end 9014 14811 a 9014 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A043553) >> /Subtype /Link H.B /ANN pdfmark end 9014 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 9549 15702 a SDict begin H.S end 9549 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A045624)p 9549 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 13388 14811 a SDict begin H.R end 13388 14811 a 13388 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A045624) >> /Subtype /Link H.B /ANN pdfmark end 13388 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 13923 15702 a SDict begin H.S end 13923 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A046088)p 13923 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 17763 14811 a SDict begin H.R end 17763 14811 a 17763 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A046088) >> /Subtype /Link H.B /ANN pdfmark end 17763 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 18297 15702 a SDict begin H.S end 18297 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A046089)p 18297 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 22137 14811 a SDict begin H.R end 22137 14811 a 22137 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A046089) >> /Subtype /Link H.B /ANN pdfmark end 22137 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 22672 15702 a SDict begin H.S end 22672 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A048882)p 22672 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 26511 14811 a SDict begin H.R end 26511 14811 a 26511 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A048882) >> /Subtype /Link H.B /ANN pdfmark end 26511 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 27046 15702 a SDict begin H.S end 27046 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A048965)p 27046 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 30885 14811 a SDict begin H.R end 30885 14811 a 30885 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A048965) >> /Subtype /Link H.B /ANN pdfmark end 30885 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 31420 15702 a SDict begin H.S end 31420 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A048966)p 31420 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 35260 14811 a SDict begin H.R end 35260 14811 a 35260 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A048966) >> /Subtype /Link H.B /ANN pdfmark end 35260 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 35795 15702 a SDict begin H.S end 35795 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049027)p 35795 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 39634 14811 a SDict begin H.R end 39634 14811 a 39634 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049027) >> /Subtype /Link H.B /ANN pdfmark end 39634 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 40169 15702 a SDict begin H.S end 40169 15702 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049028)p 40169 15660 3840 43 v 0 1 0 0 TeXcolorcmyk 44008 14811 a SDict begin H.R end 44008 14811 a 44008 15492 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049028) >> /Subtype /Link H.B /ANN pdfmark end 44008 15492 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 44543 15702 a SDict begin H.S end 44543 15702 a 0 1 0 0 TeXcolorcmyk -210 x 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/Subtype /Link H.B /ANN pdfmark end 4640 16820 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5174 17031 a SDict begin H.S end 5174 17031 a 0 1 0 0 TeXcolorcmyk -211 x Fa(A049120)p 5174 16988 3840 43 v 0 1 0 0 TeXcolorcmyk 9014 16139 a SDict begin H.R end 9014 16139 a 9014 16820 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049120) >> /Subtype /Link H.B /ANN pdfmark end 9014 16820 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 9549 17031 a SDict begin H.S end 9549 17031 a 0 1 0 0 TeXcolorcmyk -211 x Fa(A049213)p 9549 16988 3840 43 v 0 1 0 0 TeXcolorcmyk 13388 16139 a SDict begin H.R end 13388 16139 a 13388 16820 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049213) >> /Subtype /Link H.B /ANN pdfmark end 13388 16820 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 13923 17031 a SDict begin H.S end 13923 17031 a 0 1 0 0 TeXcolorcmyk -211 x Fa(A049223)p 13923 16988 3840 43 v 0 1 0 0 TeXcolorcmyk 17763 16139 a SDict begin H.R end 17763 16139 a 17763 16820 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049223) >> /Subtype /Link H.B /ANN pdfmark end 17763 16820 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 18297 17031 a SDict begin H.S end 18297 17031 a 0 1 0 0 TeXcolorcmyk -211 x Fa(A049224)p 18297 16988 3840 43 v 0 1 0 0 TeXcolorcmyk 22137 16139 a SDict begin H.R end 22137 16139 a 22137 16820 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049224) >> /Subtype /Link H.B /ANN pdfmark end 22137 16820 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 22672 17031 a SDict begin H.S end 22672 17031 a 0 1 0 0 TeXcolorcmyk -211 x Fa(A049323)p 22672 16988 3840 43 v 0 1 0 0 TeXcolorcmyk 26511 16139 a SDict begin H.R end 26511 16139 a 26511 16820 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049323) >> /Subtype /Link H.B /ANN pdfmark end 26511 16820 a Black Fa(,)p 0 1 0 0 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0 1 0 0 TeXcolorcmyk -211 x Fa(A049349)p 48917 16988 3840 43 v 0 1 0 0 TeXcolorcmyk 52757 16139 a SDict begin H.R end 52757 16139 a 52757 16820 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049349) >> /Subtype /Link H.B /ANN pdfmark end 52757 16820 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 800 18359 a SDict begin H.S end 800 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049350)p 800 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 4640 17468 a SDict begin H.R end 4640 17468 a 4640 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049350) >> /Subtype /Link H.B /ANN pdfmark end 4640 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5174 18359 a SDict begin H.S end 5174 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049351)p 5174 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 9014 17468 a SDict begin H.R end 9014 17468 a 9014 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI 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0 0 TeXcolorcmyk 22137 17468 a SDict begin H.R end 22137 17468 a 22137 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049375) >> /Subtype /Link H.B /ANN pdfmark end 22137 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 22672 18359 a SDict begin H.S end 22672 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049376)p 22672 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 26511 17468 a SDict begin H.R end 26511 17468 a 26511 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049376) >> /Subtype /Link H.B /ANN pdfmark end 26511 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 27046 18359 a SDict begin H.S end 27046 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049377)p 27046 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 30885 17468 a SDict begin H.R end 30885 17468 a 30885 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049377) >> /Subtype /Link H.B /ANN pdfmark end 30885 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 31420 18359 a SDict begin H.S end 31420 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049378)p 31420 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 35260 17468 a SDict begin H.R end 35260 17468 a 35260 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049378) >> /Subtype /Link H.B /ANN pdfmark end 35260 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 35795 18359 a SDict begin H.S end 35795 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049385)p 35795 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 39634 17468 a SDict begin H.R end 39634 17468 a 39634 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049385) >> /Subtype /Link H.B /ANN pdfmark end 39634 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 40169 18359 a SDict begin H.S end 40169 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049402)p 40169 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 44008 17468 a SDict begin H.R end 44008 17468 a 44008 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049402) >> /Subtype /Link H.B /ANN pdfmark end 44008 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 44543 18359 a SDict begin H.S end 44543 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049403)p 44543 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 48383 17468 a SDict begin H.R end 48383 17468 a 48383 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049403) >> /Subtype /Link H.B /ANN pdfmark end 48383 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 48917 18359 a SDict begin H.S end 48917 18359 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049404)p 48917 18317 3840 43 v 0 1 0 0 TeXcolorcmyk 52757 17468 a SDict begin H.R end 52757 17468 a 52757 18149 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049404) >> /Subtype /Link H.B /ANN pdfmark end 52757 18149 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 800 19687 a SDict begin H.S end 800 19687 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049410)p 800 19645 3840 43 v 0 1 0 0 TeXcolorcmyk 4640 18796 a SDict begin H.R end 4640 18796 a 4640 19477 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049410) >> /Subtype /Link H.B /ANN pdfmark end 4640 19477 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 5265 19687 a SDict begin H.S end 5265 19687 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049411)p 5265 19645 3840 43 v 0 1 0 0 TeXcolorcmyk 9105 18796 a SDict begin H.R end 9105 18796 a 9105 19477 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://oeis.org/A049411) >> /Subtype /Link H.B /ANN pdfmark end 9105 19477 a Black Fa(,)p 0 1 0 0 TeXcolorcmyk 9731 19687 a SDict begin H.S end 9731 19687 a 0 1 0 0 TeXcolorcmyk -210 x Fa(A049412)p 9731 19645 3840 43 v 0 1 0 0 TeXcolorcmyk 13570 18796 a SDict begin H.R end 13570 18796 a 13570 19477 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] 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