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 [MATH]n\in N[/MATH], consider:
[MATH]S_n=\sum_{k=0}^n (-1)^k\binom{n}{k}k^n[/MATH]
Show:
[MATH]S_n=-nS_{n-1}+n\sum_{k=0}^n (-1)^k\binom{n}{k}k^{n-1},\quad n\ge1[/MATH]
Any helps is greatly appreciated!
For each [MATH]n\in N[/MATH], consider:
[MATH]S_n=\sum_{k=0}^n (-1)^k\binom{n}{k}k^n[/MATH]
Show:
[MATH]S_n=-nS_{n-1}+n\sum_{k=0}^n (-1)^k\binom{n}{k}k^{n-1},\quad n\ge1[/MATH]
Any helps is greatly appreciated!