Catalan Number for unequal $m$ and $n$

Question

The Catalan numbers can be interpreted as the number of paths on an $n\times n$ grid that are never above the diagonal.

I am trying to figure out the generalization where the paths are on an $n\times m$ grid, $m>n$, and the paths do not cross the $(0, 0)--(n, n)$ diagonal.

Any hints?

Edit after 4 years: this is https://en.wikipedia.org/wiki/Bertrand%27s_ballot_theorem

Answer 1

I don't think they have a well established and unambiguous name but Ira Gessel and Guoce Xin have some related work in the following two papers:

• Ira M. Gessel, Super ballot numbers [PDF], J. Symbolic Computation 14 (1992), 179–194.

• I. M. Gessel and G. Xin, A combinatorial interpretation of the numbers $6(2n)!/n!(n+2)!$ [PDF], J. Integer Seq. 8 (2005) Article 05.2.3