Dyck Path III

A Dyck Path of length $2n$ in the plane is a path from $(0,0)$ to $(n,n)$ in the plane where the path never crosses (but can touch) the $y=x$ line and the only possible movements from $(x,y)$ are $(x+1,y)$ and $(x,y+1)$ . Let $N(n,k)$ be the number of length $2n$ Dyck paths that touch the diagonal $y = x$ exactly $0 \leq k \leq n$ times, excluding the initial point $(0,0)$ . By convention, define $N(0,0) = 1$ and $N(n,0) = 0$ for $n > 0$ . Find an expression for $N(n,k)$ and evaluate it when $n = 7$ and $k = 3$ .

Answer format: enter an integer.