It is well-known that the Catalan numbers have an asymptotic approximation
$$C_n\sim \frac{4^n}{\sqrt{\pi}n^{3/2}}.$$
I am curious about combinatorial interpretations of this formula, rather than a formal derivation (for example using the Stirling approximation.) For example, why would I expect $4^n$ and $n^{3/2}$ to be in this formula?
Pick your favourite combinatorial proof of the fact that $C_n=\frac1{n+1}\binom{2n}n$ (unless that’s already how you define the Catalan numbers). The central binomial coefficient $\binom{2n}n$ counts the simple random walks that start from the origin and end up at the origin after $2n$ steps. The simple random walk spreads out as $\sqrt n$, approximating a Gaussian by the central limit theorem, and thus the probability to end up at the origin after $2n$ steps decays as $\frac1{\sqrt n}$. There are $4^n$ instantiations in all, so $\binom{2n}n\sim\frac{4^n}{\sqrt n}$.