Catalan Numbers

In this work, we define the Catalan numbers $C_n$ 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 $C_n\sim \frac{4^n}{\sqrt{\pi }{n}^{3/2}}$, 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.