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:
- $B_n = \ds \sum_{k \mathop = 1}^n {n \brace k}$
where $\ds {n \brace k}$ 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:
- $B_n = \ds \sum_{k \mathop = 0}^n {n \brace k}$
But when $n > 0$:
- $\ds {n \brace 0} = 0$
Hence the result.
$\blacksquare$