Calculate sums for all given m.

Asked Nov 22 '20 at 18:59

Active Nov 23 '20 at 06:54

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Let: $S_m = \sum_{k=0}^n{((-1)^{n-k}}C_n^kk^m)$.

I need to calculate $S_m$ for all $m=0, 1, \ldots, n-1$.

As I undertood, it should be equal to zero for all $n, m$ (according to some computations on my PC). But I have no idea how to solve it more formally.

edited Nov 23 '20 at 06:54

asked Nov 22 '20 at 18:59

Someone

@BernardPan This is nonsense, you cannot put the $k^m$ out of the sum !! – TheSilverDoe Nov 22 '20 at 19:03
@TheSilverDoe You are right. I did not notice this. – Bernard Pan Nov 22 '20 at 19:05
@Someone Do you know how to deal with the case $m=1$ ? – TheSilverDoe Nov 22 '20 at 19:07

Calculate sums for all given m.

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