Closed form for a sequence defined recursively

Question

Let $a_k$ be a sequence such that $a_0=0, a_1=0, a_2=1, a_3=1$ and $$a_{k+4}=-\frac{a_{k}+ka_{k+2}}{(k+1)(k+2)}$$ for $k\ge 0$. My question is: Is a closed form formula for $a_n, n\ge 4$ possible? This problem arises from my attempt to find the power series solution for $y''+xy'+x^2y=0$.