Solution
= Solution
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
$$
C(x)=\frac{1-\sqrt{1-4x}}{2x}.
$$