CN-TST 2011 Problem 5

Let $n$ be a positive integer and let $\alpha_n$ be the number of $1$ 's within binary representation of $n$ .

Show that for all positive integers $r$ , $$2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.$$