An Invitation to Analytic Combinatorics

This is the website for the textbook An Invitation to Analytic Combinatorics: From One to Several Variables by Stephen Melczer.

The unedited manuscript, as it was submitted, is available here

This textbook currently undergoing editing. If you notice any errors, or have any comments, please send them to the author at the email address on the title page! Also contact the author if you would like to be notified when the final, edited, version is available for purchase.

Code for Examples

The following code works through examples from the textbook. Unless otherwise specified, all worksheets are for the Maple computer algebra system. To download them you may need to right click on the corresponding link and select “Download”.

Some of the Maple worksheets in Chapter 7 and further use the Kronecker package of Melczer and Salvy, available at

The Sage worksheets use the ore_algebra package, available at


Lattice Path Examples (Sage Worksheet)

Chapter 2

Alternating Permutations

Fibonacci Numbers

Look and Say Digit Sequence

Rooted Trees

Sum of Squares

Creative Telescoping Integral

Central Binomial Coefficients Squared

ODE Analysis Example (Sage Worksheet)

Central Binomial Coefficients ODE Analysis (Sage Worksheet)

Chapter 3

Catalan Diagonal

Two Diagonal Expressions for Apéry

Drawing Amoebas

Amoeba Examples

$r$-Diagonal to Main Diagonal

Chapter 5

Code for Smooth Asymptotic Expansions

Leaves of Rooted Plane Tree Example

Chapter 6

Simple Walk Asymptotics

Chapter 7

Combinatorial Examples

Non-Combinatorial Examples

Kronecker Representation Example

Work Still to Come

Most of the exercises in Parts 1 and 2 of the textbook are currently represented, and more worksheets are being added consistently. The current work order is: (1) continue to add Maple worksheets until all textbook exercises are covered; (2) add pdf versions of the Maple worksheets to make them accessible to those without Maple; (3) add Sage versions of the Maple worksheets to make them accessible to those without Maple.