Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind/Corollary

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Theorem

Let $B_n$ be the Bell number for $n \in \Z_{> 0}$.

Then:

$B_n = \displaystyle \sum_{k \mathop = 1}^n {n \brace k}$

where $\displaystyle {n \brace k}$ denotes a Stirling number of the second kind.


Proof

From Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind, we have that:

$B_n = \displaystyle \sum_{k \mathop = 0}^n {n \brace k}$

But when $n > 0$:

$\displaystyle {n \brace 0} = 0$

Hence the result.

$\blacksquare$