Dyck path

A Dyck path of semilength $n$ starts at $(0,0)$, ends at $(2n,0)$, uses steps $(1,1)$ and $(1,-1)$, and never passes below the horizontal axis.

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