Series Solutions · Superchallenge problem

Let $y = \sum_{n=0}^{\infty} a_n x^n$ be the power-series solution of $(1 + x^2)\,y'' + 4x\,y' + 2y = 0$ with $y(0) = 1$, $y'(0) = 2$. Find the exact value of the series at $x = \dfrac12$. Express your answer as a common fraction.