Frobenius & Special Functions · Easy problem

For $2xy'' + y' - y = 0$, a Frobenius series $y = \sum_{n\ge0} a_n x^{n+r}$ leads to the recurrence $(n+r)(2n+2r-1)\,a_n = a_{n-1}$ for $n \ge 1$. Take the root $r = \tfrac12$ and $a_0 = 1$. Compute $a_2$. Express your answer as a common fraction.