Having read on Wikipedia about generating functions, I still don't understand what I can do with a generating function.
For example, if I want to compute the first spread polynomial, how can I do that from a generating function, such as these (taken from the Wikipedia article):
$$ \sum_{n=1}^\infty S_n(s)x^n = {sx(1+x) \over (1-x)^3 + 4sx(1-x)}. $$
An exponential one:
$$ \sum_{n=1}^\infty {S_n(s)\over n!} x^n = {1 \over 2} e^x \left [ 1-e^{-2sx} \cos\left (2x \sqrt{s(1-s)}\right )\right ] $$