Combinatorial Math

migvas99

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Apr 13, 2021
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I'm stuck in this question. It seems so easy, but I can't see it and at this point I spent too many time on it to be able to look at it with fresh eyes.

For each \(\displaystyle n\in N\), consider:
\(\displaystyle S_n=\sum_{k=0}^n (-1)^k\binom{n}{k}k^n\)

Show:
\(\displaystyle S_n=-nS_{n-1}+n\sum_{k=0}^n (-1)^k\binom{n}{k}k^{n-1},\quad n\ge1\)

Any helps is greatly appreciated!
 
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