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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-145/3/a/solution