In [A080936][1]  gives the number of Dyck paths of length $2n$ and height exactly $k$ and has a little more information on the generating functions.

For all $n\geq 1$ and $\frac{(n+1)}{2}\leq k\leq n$ we have: 
$$T(n,k) = \frac{2*(2*k+3)*(2*k^2+6*k+1-3*n)*(2*n)!}{((n-k)!*(n+k+3)!)}.$$

>- I couldn't find any proof for the above equality and any source (article, book, (etc,.)?

>- I need to understand how to construct generating functions and formulas The number of Dyck paths of length $2n$ and height exactly $k$.








  [1]: https://oeis.org/A080936