The ordered Bell numbers as weighted sums of odd or even Stirling numbers of the second kind

Abstract

For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity ${\sum }_{k=1}^{n/2}S(n,2k)(2k-1)!=B(n-1)$. An analogous identity holds for the sum over odd $k$'s.

3 pages