My question is rather simple. I have a recurrence relation. After many transformations, I have arrived at the point where I know the generating function is $$A(x) = \dfrac{2x}{(1-x)^4}$$ Now I want to know the series from which this function is derived, so I can get some sort of a closed formula for the series $a_n$.
Normally, when in this situation, I would perform some transformation so the function has a form which I can recognize, and from which I can find the series. However with this I am rather stuck.
(The function $A(x)$ also has other terms, which I already know the series of. It is this that is being a pain)
Hint : \begin{eqnarray*} \frac{1}{(1-x)^{n+1}} = \sum_{k=0}^{\infty} \binom{n+k}{k}x^k. \end{eqnarray*}