# 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.

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,...
2. 1,0,1,0,...
3. 1,-1,1,-1,...
4. 1,c,c^2,c^3,...
5. 1,2,3,4,...
6. 1,1/2,1/3,1/4,...
7. Fibonacci numbers
8. Harmonic numbers
9. Number of binary trees
10. Number of partition trees
• Functional equations
1. from recurrence relations (eg. Hanoi, Fibonacci, convolution and binary trees)
2. from combinatorial constructions (binary trees, partition trees)
• Solving recurrence relations using generating functions
1. partial fractions (Hanoi, Fibonacci)
2. 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