Alternating binomial moment ($S_r(n)={\sum }_{m=0}^n(-1{)}^m{m}^r(\genfrac{}{}{0ex}{}{n}{m})$)

For nonnegative integers $r,n$, the alternating binomial moment satisfies $S_r(n)=0$ when $r<n$ and $S_n(n)=(-1{)}^{n}n!$. Expand the polynomial $X^r$ in falling factorials and differentiate the binomial theorem to prove these identities. Equivalently, alternating binomial coefficients implement a finite difference that annihilates polynomials of degree less than $n$.