Question 88
Define a sequence of integers by $a_0 = 1$, and $a_n = \sum_{k=0}^{n-1} \binom{n}{k} a_k$, $n \ge 1$. Let $m$ be a positive integer, let $p$ be a prime, and let $q$ and $r$ be non-negative integers. Prove that the difference $a_{p^m q+r} - a_{p^{m-1} q+r}$ is divisible by $p^m$.