<s>[INST] Sequence corresponding to the generating function 
Find the sequence corresponding to the generating function $$G(x) = \frac{2x^4}{2x^3-x^2-2x+1}$$

First of all, I wrote this equation like that;
$\sum\limits_{n=0}^\infty(a_n)x^n = \frac{2x^4}{2x^3-x^2-2x+1}$ 
Then, I think right hand side should be;
$\frac{2x^4}{2x^3-x^2-2x+1} = \frac{A}{2x-1} + \frac{B}{x-1} + \frac{C}{x+1}$
I found 
$A = \frac{-2x^2}{3}$ , $B = x^2$ ,  $C = \frac{x^2}{3}$
However, I cannot continue after that point.
 [/INST] Assuming the partial fraction decomposition is correct (I haven't checked it, but I suppose you have), remember that $\frac{1}{1-x}=\sum_{n=0}^\infty x^n$. Hence $$\frac{1}{2x-1}=-\frac{1}{1-2x}=-\sum_{n=0}^\infty (2x)^n=-\sum_{n=0}^\infty 2^nx^n$$
and $$\frac{1}{x-1}=-\frac{1}{1-x}=-\sum_{n=0}^\infty x^n$$
and $$\frac{1}{x+1}=\frac{1}{1-(-x)}=\sum_{n=0}^\infty (-1)^nx^n$$
Remember also that multiplying a power series by $x^n$ shifts the coefficients $n$ steps upwards. Can you continue from here?
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