How would I find a closed-form solution of a recurrence that I obtained from solving an ODE using Frobenius method?:
$(k+2)(k + 4)a_{k+2} - a_{k+1} + (k+2)a_k = 0$
given $a_0 = a$ and $a_1 = \frac{1}{3}a.$ I am fairly new to the topic. I have already searched for a pattern from the first few terms (I'm terrible at guessing games), tried substituting other series, using annihilators and generating functions. Various papers I have found on the topic seem to avoid examples of variable coefficients in recurrences of higher order than $1$.
May I ask for a method I should use, or perhaps why in my case there can't be a closed-form solution? I would also appreciate a few steps into the algorithm in question.
Equation in question: $x^2y'' + x(x^2-1)y'-xy=0$
EDIT: Upon evaluating denominators of the first few terms I plugged them into OEIS, and obtained:
$a_k = \frac{2b_k}{k!(k+2)!}$
Using $b_k$ as a new substitution results in:
$b_{k+2} - b_{k+1} + b_k(k+1)(k+2)(k+3)=0$