Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.5

A Generalized Recurrence for Bell Numbers


Michael Z. Spivey
Department of Mathematics and Computer Science
University of Puget Sound
Tacoma, Washington 98416-1043
USA

Abstract:

We show that the two most well-known expressions for Bell numbers, $ \varpi_n = \sum_{k=0}^n \genfrac{\{}{\}}{0pt}{}{n}{k}$ and $ \varpi_{n+1} = \sum_{k=0}^n \binom{n}{k} \varpi_k$, are both special cases of a third expression for the Bell numbers, and we give a combinatorial proof of the latter.

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(Concerned with sequences A000110 A007318 A008275 and A008277 .)

Received January 11 2008; revised version received May 27 2008. Published in Journal of Integer Sequences, June 23 2008.

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