Journal of Integer Sequences, Vol. 15 (2012), Article 12.2.4

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


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.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequence A000110.)

Received December 17 2011; revised version received December 30 2011. Published in Journal of Integer Sequences, December 30 2011.

Return to Journal of Integer Sequences home page