Counting Generalized Dyck Paths

The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0, 0) to (n, n) which is below the diagonal line y = x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0, 0) to (m, n) ∈ N2 which is below the diagonal line y = n m x, and denote by C(m, n) the number of Dyck paths from (0, 0) to (m, n). In this paper, we give a formula to calculate C(m, n) for arbitrary m and n.