Let $C_{k,M}$ be the number of Dyck paths of length $2k$ that stay $\leq M$. For $M= \infty$ we have $C_{k,\infty} \approx 4^k / k^{3/2}$. I'm wondering if we can get arbitrarily close to this growth rate for $M$ that does not depend on $k$.
More precisely, is it true that for any fixed $\delta >0$ there exists $M = M(\delta)$ such that
$$\frac{C_{k,M}}{(4 -\delta)^k} \to \infty?$$
Or does $M$ also need to grow with $k$ for this to hold?
This is true for $M= M(\delta)$ independent of $k$. Let $C_M$ be the number of Dyck paths of length $2M$ with no height restrictions. One way to stay inside the range $[0,M]$ is to concatenate $k/M$ Dyck paths of length $2M$. Thus, $$C_{k,M} \geq (C_M)^{k/M} \approx \left( \frac{4^M}{M^{3/2}} \right)^{k/M} = \left(\frac{4}{M^{3/(2M)} }\right)^k.$$
Since $M^{a/M} \to 1$ as $M \to \infty$ for any $a >0$, we can choose $M$ large enough so that $M^{3/(2M)} > 1- \epsilon.$ We then have $$C_{k,M} \geq (4/(1-\epsilon))^k > (4-\delta)^k$$ for $\epsilon$ small enough.