The Dual of Spivey's Bell Number Formula
István Mező
Department of Applied Mathematics and Probability Theory
Faculty of Informatics
University of Debrecen
H-4010 Debrecen
P. O. Box 12
Hungary
Abstract:
We give the dual of Spivey's recent formula for Bell
numbers. The dual involves factorials and Stirling numbers of the first
kind. We point out that Spivey's formula immediately yields the famous
Touchard congruence for Bell numbers. Finally, we extend these formulas
to the r-Stirling case.
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(Concerned with sequence
A000110.)
Received December 17 2011;
revised version received December 30 2011.
Published in Journal of Integer Sequences, December 30 2011.
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