I am trying to solve this problem, where I am given a recurrence relation, for which I need to find all solutions.
The recurrence relation is as follows: $a_n - 2na_{n-1} + n(n-1)a_{n-2} = 2nn!$, with initial conditions $a_0$ and $a_1$ = 1, and $n \geq 2$.
Using exponential generating functions, I have concluded that the generating function $A(x)$ of this recurrence relation, where $$A(x) = \sum_{n\geq 0} a_n \dfrac{x^n}{n!},$$ is given by $$A(x) = \dfrac{2x-3x(1-x)^2 + (1-x)^2}{(1-x)^4}$$.
How would I continue? I cannot see a way to transform this formula into something I can recognize? Normally when I reach this point I try to transform my function into something like one of these, from which I can then find my series.