What is the number of (general?) Dyck paths from $(0,0)$ to $(2n,k_1)$, where $k_1\geq0$, allowing the path to go below the $x$ axis and touch the negative horizontal line at $k_2\leq0$ an arbitrary number of times?
It seems to be equivalent to the number of paths starting at $(x_1,y_1)$ and ending at $(x_2,y_2)$ (all positive), where the paths are not allowed to touch the $x$ axis. Is it correct?
Wonder why nobody answered but the result is $$ \binom{x_2}{{1\over2}(x_2+k_1)}-\binom{x_2}{{1\over2}(x_2-k_1+2k_2)}, $$ where $x_2=2n$ but in this form $x_2$ can be odd too. It is an equivalent to the Bertrand's ballot problem -- just offset to the lattice origin.