Series Solutions · Superchallenge problem

Let $y = \sum_{n=0}^{\infty} a_n x^n$ be the power-series solution of $y'' = (1 + x^2)\,y$ with $y(0) = 1$, $y'(0) = 0$. Find the exact value of $\sum_{n=0}^{\infty} a_n$.