How to calculate a combinatorial sum I have a combinatorial sum in hand which I suspect equals zero. But I do not know how to prove it. Can you guys help me? (I am even not sure if this is a hard question or not)
Is
$$
\sum_{k=0}^n (-1)^k k^{n-1} \left(\begin{array}{l}
n \\
k
\end{array}\right) = 0
$$
?

HINT: Start with $$f(x)=(1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k.$$ Differentiate to show that $f'(-1)=0$. Differentiate again to show that $f''(-1)=0$. Keep going (inductively) until the $(n-1)$-th derivative.
