Frobenius & Special Functions · Superchallenge problem

For integers $n \ge 0$ the Frobenius solution of Bessel's equation of order $n$ is $J_n(x) = \displaystyle\sum_{k\ge0}\frac{(-1)^k}{k!\,(n+k)!}\left(\frac x2\right)^{2k+n}$, and with $J_{-n} = (-1)^nJ_n$ these satisfy the generating-function identity $e^{\frac x2\left(t - \frac1t\right)} = \displaystyle\sum_{n=-\infty}^{\infty}J_n(x)\,t^n$. Compute $\displaystyle\sum_{n\ge1}n^2\,J_n(3)^2$. Express your answer as a common fraction.