Working with Generating Functions

Abstract

We have explored a range of enumerative problems by finding a recurrence relation that the corresponding count satisfies. This means that once we have some initial terms, we can progressively calculate more. We have also developed a way to convert some recurrence relations into a generating function. If we can manipulate the generating function and then expand it as a power series that enables us to give explicit expressions for the count by reading off the coefficient of a particular power of the indeterminate used. This last step is the focus of this chapter. Along the way we also explore the different recurrences satisfied by the corresponding sequence.