October Lectures

October 1

• Introduction to exponential generating functions. Examples.
• Comments on other generating functions.
• Real asymptotics; asymptotic expansions; Taylor expansions. Examples.

October 8

• Manipulation rules for asymptotic expansions.
• Bootstrapping.
• Euler-Maclaurin formula; properties of Bernoulli numbers.

October 15

• Examples of the use of the Euler-Maclaurin formula:
  • Harmonic numbers;
  • Stirling's approximation;
  • Catalan numbers.

October 22

• The Symbolic Method: basic definitions and constructions.
• The admissibility theorem for unlabelled constructions and
  ordinary generating functions.

October 29

• Examples of the use of the symbolic method:
  • Binary trees: internal, external, and total number of nodes;
  • General trees and forests;
  • Polynomials over finite fields.