This is part 2 of our series on generating functions. In part 1 we used generating functions to solve the quicksort recurrence, which is a serious industrial-strength application. This time we will use them for something a lot more whimsical: counting how many different cans of colored marbles you can put together under a bunch of arbitrary rules.

The example is from Trotter's Applied Combinatorics. Our working notes for this one, along with a few related exercises, live on the Generating Functions page of our wiki.

The Problem

We are packing cans of marbles. Each can holds 20 marbles, in …