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The Catalan number
$$
C_n=\frac1{n+1}\binom{2n}{n}
$$
counts <Dyck paths> of semilength $n$. Its ordinary generating function $C(x)=\sum_{n\geq0}C_nx^n$ satisfies $C(x)=1+xC(x)^2$.

Catalan number

A nonempty <Dyck path> decomposes uniquely as an up-step, a Dyck path, a down-step, and another Dyck path. Marking each matched outer pair by $x$ gives
$$
C(x)=1+xC(x)^2.
$$
The solution with constant term one is
$$
C(x)=\frac{1-\sqrt{1-4x}}{2x}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 3 / a / Solution

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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.


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 Incoming links: Dyck path