I've been trying to find the number of Dyck paths $P$ of length $2n$ such that $\forall (x,y) \in P, |x-y| \le k$ for some fixed constant $k$. These are the Dyck paths that are bounded by the lines $y=x$, $y=x-k$, $y=0$, and $x=n$. This is also the number of trapezoidal parallelogram polyominoes.
If we let $P(n,k)$ be the number of paths, it is easy to prove that $C_n \ge P(n,k) \ge (C_k)^{n/k}$, where the first equality is tight if $n\le k$ and the final equality is tight only for $k=1$.
This question may be too general, but does anyone know of a closed form for the function $P(n,k)$? Or at least have a clue about how to continue towards one?
Counting Dyck paths can be rephrased as the problem of counting walks on the semi-infinite path graph $\mathbb{Z}_{\ge 0}$ from the origin $0$ to itself. Counting these restricted paths is equivalent to the problem of counting walks on this graph which do not stray more than $k$ from the origin, which is equivalent to the problem of counting walks on a finite path graph of length $k$ from one end to itself.
For fixed $k$ this sequence is described by a linear recurrence, or equivalently it has rational generating function. These generating functions are written down somewhat explicitly in this blog post: they appear as convergents of a continued fraction
$$\frac{1}{1 - \frac{x}{1 - \frac{x}{1 - ...}}}$$
describing the generating function of the Catalan numbers.