Frobenius & Special Functions · Challenge problem

Bessel's equation of order $0$, $xy\,\prime\prime + y\,\prime + xy = 0$, has the solution $J_0(x) = \sum_{k\ge0}\dfrac{(-1)^k}{(k!)^2}\left(\dfrac x2\right)^{2k}$. Compute the coefficient of $x^4$. Express as a common fraction.