The number of dyck path of height less than or equal to $h$ Heyllo guys!
I was wondering whether there were closed form, recurrence relation, or generating function for the number of dyck path of height less than or equal to $h$; the height of dyck path is equal to the highest $y$ value that is touched by the path.
Thank you for any help!. Hope you guys have a nice day.
 A: There seems to be a proof in this paper that the generating function for the number of Dyck paths bounded by height $k$ is given by $E_k=\frac{F_k}{F_{k+1}}$, where $F_k$ are determined by the recurrence relation $F_0=F_1=1$, $F_k=F_{k-1}-t^2 F_{k-2}$. 
Besides that, here is the entry in the OEIS, as mentioned in the answer to this question (the same question as yours). In it, an explicit formula for $T(n,k)$ - being the number of Dyck paths bounded in height by $k$ of length $2n$ - appears: 
\begin{equation*}
T(n,k)=\frac{2^{2n+1}}{k+2}\sum_{i=1}^{k+1} \left[\sin\left(\frac{\pi i}{k+2}\right)\cos\left(\frac{\pi i}{k+2}\right)^n\right]^2.
\end{equation*}
Now I'm not sure that this formula is correct, but it satisfies the initial conditions and "converges" to the Catalan numbers as $k\to\infty$ (or at least I've checked the first 100 numbers using Mathematica). One would have to check that it satisfies the correct recurrence relation
\begin{equation*}
T(n+1,k) = \sum_{i=0}^{n} T\left(i,k\right)T\left(n-i,k-1\right)
\end{equation*} 
as mentioned in Brian M. Scott's answer to the question mentioned before. 
