# Lecture 9: Generating Functions

This lecture continues the theme of using Maple to investigate the performance
of algorithms and generating combinatorial objects. The lecture introduces
generating functions which are a very important tool for analyzing integer
sequences, recurrence relations, and combinatorial objects.

### Background Material

- Recursion and recurrence relations.
- Power series.
- Taylor series.

### Reading

Review the material on generating combinatorial objects in
Also study Maple's series command.
### Topics

- Introduction to generating functions (simple examples and elementary properties)
- 1,1,1,...
- 1,0,1,0,...
- 1,-1,1,-1,...
- 1,c,c^2,c^3,...
- 1,2,3,4,...
- 1,1/2,1/3,1/4,...
- Fibonacci numbers
- Harmonic numbers
- Number of binary trees
- Number of partition trees

- Functional equations
- from recurrence relations (eg. Hanoi, Fibonacci, convolution and
binary trees)
- from combinatorial constructions (binary trees, partition trees)

- Solving recurrence relations using generating functions
- partial fractions (Hanoi, Fibonacci)
- binomial theorem (number of binary trees)

### Maple worksheets and programs and other resources

- gen.mw - Maple worksheet on generating functions.
computation

### Assignments

Created: Oct. 26, 2006 by jjohnson@cs.drexel.edu