I am interested in counting the following. How many words using $n-1$ copies of $u$ and ${n \choose 2} - n+1$ copies of $d$ begin with $uu$ and, in general, the $k^{th}$ $u$ is among the first ${k \choose 2} + 1$ letters in the word?
This is a generalization of a Dyck path. I need to start at the origin, end at $(n-1,2(n-1)-{n \choose 2})$, and always stay above the corresponding curve.
I'm looking for a reference or an argument that gives a formula analogous to the Catalan numbers. I can write down the answer as a messy sum.
Thanks!