Find seqeunce that is reated to genereating function

Question

Is there any way to find a sequence from a generating function? As an example how can I find sequence that is treated to generating function of $g(x)= \frac{x+1}{(1-x)^3}?$

Answer 1

Note that $$ \frac{1}{(1-x)^3}=\sum_{n=0}^{\infty}\binom{n+2}{2}x^n $$ by differentiating the identity $$ \frac{1}{(1-x)}=\sum_{n=0}^{\infty}x^n $$ three times. Thus $$ \begin{align} \frac{1}{(1-x)^3}+\frac{x}{(1-x)^3} &=\sum_{n=0}^{\infty}\binom{n+2}{2}x^n + \sum_{n=1}^{\infty}\binom{n+1}{2}x^n\\ &=1+ \sum_{n=1}^{\infty}\left[\binom{n+2}{2}+\binom{n+1}{2}\right]x^n \end{align} $$

Answer 2

Make a Taylor series expansion around $x=0$. You will have

$g(x) = \sum_{i=0}^\infty a_ix^i$, where $a_i$ is the sequence you have generated by $g(x)$.