A natural extension of Catalan numbers

A Dyck path is a lattice path in the plane integer lattice ℤ × ℤ consisting of steps (1,1) and (1,-1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0,0) and finish at (2n, 0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p₁ and P₂, for the number of Dyck paths starting at pi, ending at P₂, and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.