Frobenius & Special Functions · Challenge problem

The equation $2x^2y'' + x(1+2x)y' - y = 0$ has indicial roots $1$ and $-\tfrac12$. Let $y = x(1+a_1x+a_2x^2+\cdots)$ be the Frobenius solution for the larger root $r=1$, normalised by $a_0=1$. Find $a_1 + a_2$. Express your answer as a common fraction.