Frobenius & Special Functions · Medium problem

For Bessel's equation of order $\tfrac12$, $x^2y'' + xy' + \left(x^2 - \tfrac14\right)y = 0$, the Frobenius recurrence is $\left[(n+r)^2 - \tfrac14\right]a_n = -a_{n-2}$ (with $a_1 = 0$). Take $r = \tfrac12$ and $a_0 = 1$. Compute $a_4$. Express your answer as a common fraction.