Define the forward difference operator $$\Delta f(x) = f(x+1) - f(x)$$ I believe that if $f(x)$ is a polynomial with integer coefficients, $\Delta^k f(x)$ is divisible by k!. By linearity it suffices to consider a single monomial $f(x) = x^n$. I've checked this for small values of $n$ and $k$, and believe that a simple proof should exist, but am unable to find it.

In particular, brute force gives $$\Delta^k x^n = \sum_j x^{n-j} \left[ \binom{n}{j} \sum_i (-1)^{k-i} \binom{k}{i} i^j \right]$$ but the terms in brackets appear to have no closed form solution (see (20)-(25) of http://mathworld.wolfram.com/BinomialSums.html).

Motivation: I have an unknown integer coefficient polynomial of degree $n$ sampled at $x = 0, 1, \ldots, n$, and want to prove that all intermediate results in the classical divided difference algorithm are integers.