## Series Solution to Ordinary Differential Equation Recurrence Relation

The original question is:

$(4+x^2)y'' + 3xy' + y = 0$

The guessed solution is $y = \sum_{n=0}^\infty a_n x^n$After some calculation, I have reduced everything to a single infinite sum plus some terms involving some terms of the sequence.

As for my recurrence relation for the sequence, here are the equations I have found:

$8a_2 + a_0 = 0$

$24a_3 + 4a_1 = 0$

$a_n = \frac {4(n+2)a_{n+2}}{n+1}
$

So what do I do now to find the general formula for the sequence of coefficients?