Generating Functions For Partitions
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Herbert Wilf says that "a generating function is a clothesline on which we 
hang up a sequence of numbers for display."  Less fancifully, it's just a 
power series which we manipulate to carry out some desired operation on 
its coefficients.  It's very useful for counting things, especially things 
made by putting together smaller things.  I'll illustrate this with the 
generating functions that count partitions of natural numbers.  We'll find 
proofs of such cheerful facts as: the number of partitions of n into parts 
of distinct size equals the number of partitions of n into parts of odd 
size.