Frobenius & Special Functions · Medium problem

The equation $4xy'' + 2y' + y = 0$ has indicial roots $0$ and $\tfrac12$ at $x = 0$. Let $y = \sum_{n\ge0}a_nx^n$ be the Frobenius solution for $r = 0$ with $a_0 = 1$. Sum the series in closed form and compute $y\!\left(\dfrac{\pi^2}{9}\right)$. Express your answer as a common fraction.