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kodlu
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The number of Dyck paths of length $2n$ and height exactly $k$

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 $k$.
1Spectre1
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