I am new to generating functions but understand how to derive them from given discrete numeric functions. Is there a simple way to derive the discrete numeric function given a generating function. For example, given
$$ a_r = 3^r,\quad \text{where}\ r = 0, 1, 2, \ldots, $$ its generating function is $$ A(z) = \frac{1}{1-3z}. $$ I want to know if there is a way to get $a$ given a generating function without listing its closed form as a series and gathering the coefficients of terms.
On
Nope, but... Roman's "Umbral Calculus" has a bunch of relations that could be used to find Sheffer Sequences: i.e. recurrences that satisfy them.
$g(t)\cdot e^{x\cdot f(t)}=a_{0}+a_{1}\left(x\right)\cdot t+a_{2}\left(x\right)\cdot t^{2}\cdots$
In his case, one would usually start with x=0 or x=1. Testing his relations against your particular problem. His Chapter 3, section 7 gives iff Theorems.
https://www.amazon.com/Umbral-Calculus-Steven-Roman/dp/0486441393
A lot of common series are of this nature.
In general, there is no way to get a closed form for the term from the generating function, and viceversa. Many, many "simple" generating functions have very complex coefficients, or are even used to define new sequences (for example, the generating function $\frac{z}{e^z - 1}$ defines the Bernoulli numbers). Some simple cases can be resolved in a systematic way, if the generating function is a rational function (a fraction of polynomials) then splitting into partial fractions gives geometric series and other series that can be handled by the (generalized) binomial theorem.