Frobenius & Special Functions · Hard problem

Laguerre's equation $xy'' + (1 - x)y' + 3y = 0$ has a double indicial root $r = 0$ at $x = 0$, and its Frobenius solution $y = \sum_{k\ge0}a_kx^k$ with $a_0 = 1$ terminates. Compute $\displaystyle\int_0^\infty e^{-x}\,x^3\,y(x)\,dx$.