Counting peaks at height k in a Dyck path

A Dyck path is a lattice path in the plane integer lattice ℤ × ℤ consisting of steps (1, 1) and (1, -1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y = k that is immediately preceded by a (1, 1) step and immediately followed by a (1, -1) step. In this paper we find an explicit expression for the generating function for the number of Dyck paths starting at (0, 0) and ending at (2n, 0) with exactly r peaks at height k. This allows us to express this function via Chebyshev polynomials of the second kind and the generating function for the Catalan numbers.