Lecture Videos Lecture 19 – Generating Functions

Lecture 19 – Generating Functions

You may want to download the lecture slides that were used for these videos (PDF).

1. Deceiving Infinite Series

This video introduces a geometric series problem that motivates our discussion in subsequent videos on generating functions. (6:05)

2. Generating Functions – Introduction

This video describes the idea behind a generating function. (1:23)

3. Generating Functions – Examples

A number of examles of generating functions are explored in this video. (9:35)

4. Generating Functions – Binomial Coefficients

This video discusses the generating function for an old friend, the sequence an = C(n+r-1,r-1). (3:30)

5. Partitions of an Integer

This video introduces the concept of a partition of integer n, and discusses the partition of the integer 6. (4:57)

6. Partitions of the Integer 7

This video discsses the partition of the integer 7. (5:43)

7. Partitions into Distinct Parts

We want to prove that for every positive integer n, the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. We will complete this proof with generating functions. Here, we find a generating function for the number of partitions of n into distinct parts. (5:55)

8. Partitions into Odd Parts

We introduce the generating function g(x), whose nth coefficient bn is the number of partitions of the integer n into odd parts. Then, we explore examples of other generating functions. (6:50)

9. Completing Our Proof

We use the generating functions above to complete our proof that for every positive integer n, the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. (6:17)

10. Determining the Number of Partitions of a Large Integer

If we needed to determine the number of partitions for 2339745007313, could we do it? (3:08)

11. An Incredible Identity

There is a way to find bijection that maps between the partitions into distinct parts and the partitions into odd parts. This bijection is not covered in the lectures, but you should be able to find it on the Internet. (1:18)

12. The nth Term in a Taylor Expansion

This video reviews the formula for the Taylor expansion, so that we can get to the formula for the nth term of the expansion. We are interested in the case where c = 0. (2:36)

13. f(x) = (1 – 4x)-1/2

Here, we find that (1 – 4x)-1/2  is the generating function for C(2n, n). (10:49)

14. Identities Found via the Magic of Generating Functions

What is the generating function for 4n? In this video, we find an identity for 4n by combining the generating function from the previous video and geometric series. (7:48)