# Catalan Numbers

 Title: Catalan Numbers Author: Manuel Eberl Submission date: 2016-06-21 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. BibTeX: @article{Catalan_Numbers-AFP, author = {Manuel Eberl}, title = {Catalan Numbers}, journal = {Archive of Formal Proofs}, month = jun, year = 2016, note = {\url{https://isa-afp.org/entries/Catalan_Numbers.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: Landau_Symbols Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.