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Dobiński’s formula

Posted: Sat Aug 12, 2017 4:29 pm
Let $n \in \mathbb{N}$ and $\mathcal{B}_n$ denote the $n$ - th Bell number. Prove that

$$\sum_{k=0}^{\infty} \frac{k^n}{k!}=\mathcal{B}_n \cdot e$$