 Catalan number

The Catalan numbercounts Dyck paths of semilength $n$. Its ordinary generating function $C(x)={\sum }_{n\ge 0}{C}_n{x}^n$ satisfies $C(x)=1+xC(x{)}^2$.

Catalan numbers are a sequence of natural numbers that have many applications in combinatorial mathematics. The \( n \)-th Catalan number \( C_n \) can be defined using the following formula: \[ C_n = \frac{1}{n + 1} \binom{2n}{n} = \frac{(2n)!}{(n + 1)!n!