Alternating binomial moment
= Alternating binomial moment
{title2=$S_r(n)=\sum_{m=0}^n(-1)^m m^r\binom nm$}
For nonnegative integers $r,n$, the alternating binomial moment satisfies $S_r(n)=0$ when $r<n$ and $S_n(n)=(-1)^nn!$. 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 $n$.