Lecture 16: Recurrence equations, generating functions, and equivalence relations

Taught by ArsDigita
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Lesson Summary:

In this lesson, students learn how to use generating functions to solve counting problems, specifically recurrence equations. The lecture begins with a discussion on different phobias and then transitions to a review of previous lectures before introducing generating functions. The lecture then goes into detail on the Fibonacci numbers, using them as an example to demonstrate how to solve recurrence equations using generating functions. The goal is to get a closed form for f of x, which represents the coefficients in the generating function.

Lesson Description:

• How do generating functions help solve recurrence equations?
• How do you find the closed form for Fibonacci numbers using a generating function?
• How many binary trees are there with n nodes?
• How many ways are there of placing n pairs of balanced parentheses?
• How many ways are there of associating n+1 square matrices in order to multiply them?
• Why are the number of binary trees, the number of ways of placing balanced parentheses, and the number of ways of associating square matrices equivalent or the same problem, which are Catalan numbers?
• What are equivalence relations?
• How are Taylor series used with generating functions in solving counting problems?
• Staff Review

• Currently 4.0/5 Stars.
Generating functions are explained some more in this lesson. A lot of Algebra is done and many different ideas are brought together in getting a generating function for the closed form of the Fibonacci sequence. Some of the math gets really ugly. Also, three equivalent problems are shown to be the same. Taylor series also make an appearance in this lesson.