How to reconstruct distribution from the generating function

Question

Suppose $$F(x)=\sum_n p(n)x^n$$ is a generating function, and we have the expression for $F(x)$ explicitly. Then how we can get the expression for $p(n)$ from this generating function?

Answer 1

Assuming $p(n)=0$ for $n<0$,

$$p(n) = \frac{1}{n!}\left.\frac{d^n}{dx^n}F(x)\right|_{x=0}.$$

If your $F(x)=Ae^{Bx}x^C$, for $C$ a positive integer, then:

$$F(x)=A\sum_{n\geq 0} \frac{B^nx^{n+C}}{n!},$$

giving

$$p(n)=A\frac{B^{n-C}}{(n-C)!},$$

for $n\geq C$ and $p(n)=0$ otherwise.