Ordinary generating functions
Remark
the generating function is not actually regarded as a functions,it is a formal power series,so it is not required to converge
if we are given a genreating function,how can we get its corresponding sequence,the idea is to use the geometric series
$$ \frac{1}{1 - x} = \sum_{n \ge 0}x^n $$the coefficient of the $x^n$-$term$ is $[x_n]G(x) = a_1 b_1^n + a_2 b_2^n + \dots + a_k b_k^n$
Generally, we may use the Taylor expansion of $G(x)$ at $x=0$,Recall that the Taylor series of a function $G(x)$ is
$$ G(x)=\sum_{n\ge 0}\frac{G^{(n)}(0)}{n!}x^n $$Comparing the two expressions, we obtain
$$ a_n=\frac{G^{(n)}(0)}{n!} $$Hence, the coefficients of the generating function can be recovered from the derivatives of $G(x)$.
Operating on generating function
Translating sequences into polynomials is advantegeous