ELMO SL 2011 A6

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$ , $$\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),$$where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$ .

Calvin Deng.