Sequence corresponding to the generating function

Question

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.

Answer 1

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?