Frobenius & Special Functions · Hard problem

The hypergeometric equation $x(1 - x)y'' + (2 - 3x)y' - y = 0$ has indicial roots $0$ and $-1$ 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 $y\!\left(\tfrac12\right)$.