[jstrieb]

Generating functions are definitely strange and unintuitive, but they can be really helpful for combinatorics.

The introduction to generating functions in this well known text on the subject (at the beginning of Chapter 1) might be more helpful than the Wikipedia article linked in the original post:

https://www2.math.upenn.edu/~wilf/gfology2.pdf

From that text:

> Although giving a simple formula for the members of the sequence may be out of the question, we might be able to give a simple formula for the sum of a power series, whose coefficients are the sequence that we’re looking for.