In A080936 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: $$T(n,k) = \frac{2(2k+3)(2k^2+6k+1-3n)(2n)!}{((n-k)!(n+k+3)!)}.$$

• 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 $k$.