Question 168
Given a non-negative integer $n$, let $\sum_{k=0}^{2n} a_k X^k$ be the standard power expansion of the polynomial $\sum_{k=0}^{n} \binom{n}{k}^2 (X+1)^{2k} (X-1)^{2(n-k)}$. The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X \mapsto -X$. Show that the $a_{2k}$ are all positive.