I was trying to find the general term of the sequence $ a_n=\sum\limits_{k=0}^n(-1)^k \binom{n}{k}k^n$ .

When I type the following in MMA

`Sum[(-1)^k Binomial[n, k] k^n, {k, 0, n}] `

I get a simple expression $ (-1)^n n!$ .

But I find it difficult to prove this equality by induction.

So I wonder

if there exists some ways to get a step-by-step evaluation of the function.`Sum`

Note that the methods mentioned here are not working in this case.