# Generating function of recurrence sequence

I have the following recurrence sequence $$a_{1} = 0\,,\quad a_{2} = 1\,, \qquad\qquad a_{n} = {2 + 2\left(n - 2\right)\, a_{n - 2} + \left(n - 2\right)\left(n - 1\right)\, a_{n - 1} \over n\left(n - 1\right)}$$

It starts form $\displaystyle{% \left\lbrace% 0,\ 1,\ {2 \over 3},\ {5 \over 6},\ {4 \over 5},\ {37 \over 45},\ {52 \over 63},\ {349 \over 420},\ {338 \over 405},\ {11873 \over 14175} \right\rbrace }$.

Hint: Consider $A(x)=\sum\limits_{n=1}^\infty a_nx^n$ the generating function of $(a_n)$. The recursion you are interested in is that, for every $n\geqslant2$, $$n(n-1)a_n=2+2(n-2)a_{n-2}+(n-2)(n-1)a_{n-1}.$$ Can you identify the generating function of $(b_n)$ when $b_n=n(n-1)a_n$? When $b_n=2$? When $b_n=2(n-2)a_{n-2}$? And when $b_n=(n-2)(n-1)a_{n-1}$? Then see what comes out.