Generating functions are familiar from their use in finding the averages of distributions, but they can do so much more. To convince you, I can't do better than copy a short list from Herbert Wilf's excellent book 'generatingfunctionology'. Suppose you have a sequence of numbers and you want to know something about that sequence, here are some of the things you'll often be able to do with generating functions:
(a) Find an exact formula for the members of your sequence. (b) Find a recurrence formula. (c) Find averages and statistical properties of your sequence. (d) Find asymptotic formulas for your sequence. (e) Prove identities.
And even more. Guided by Wilf's book, I'll take a look at some aspects of (a), (b) and (d).
This is a roughly weekly series of didactical blackboard talks focussing on some theoretical aspect of Condensed Matter, Biological, and Statistical Physics..