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?