### Abstract

In this work, we define the Catalan numbers Cn and prove several equivalent definitions (including some closed-form formulae). We also show one of their applications (counting the number of binary trees of size n), prove the asymptotic growth approximation Cn ∼ 4n / (√π · n1.5), and provide reasonably efficient executable code to compute them.

The derivation of the closed-form formulae uses algebraic manipulations of the ordinary generating function of the Catalan numbers, and the asymptotic approximation is then done using generalised binomial coefficients and the Gamma function. Thanks to these highly non-elementary mathematical tools, the proofs are very short and simple.

### License

BSD License

### Topics

- Mathematics/Combinatorics

### Session Catalan_Numbers

- Catalan_Auxiliary_Integral
- Catalan_Numbers

### Depends on

- Landau Symbols

## Cite

×

Catalan_Numbers-AFP

    @article{Catalan_Numbers-AFP,
      author  = {Manuel Eberl},
      title   = {Catalan Numbers},
      journal = {Archive of Formal Proofs},
      month   = {June},
      year    = {2016},
      note    = {\url{https://isa-afp.org/entries/Catalan_Numbers.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

Download

## Download

× Download latest

Older releases:

- Sep 13, 2023 : Isabelle2023
- Oct 27, 2022 : Isabelle2022
- Dec 14, 2021 : Isabelle2021-1
- Feb 23, 2021 : Isabelle2021
- Apr 18, 2020 : Isabelle2020
- Jun 11, 2019 : Isabelle2019
- Aug 16, 2018 : Isabelle2018
- Oct 10, 2017 : Isabelle2017
- Dec 17, 2016 : Isabelle2016-1