Enumeration of Fuss-Schröder paths

The Hessenberg matrices and Catalan and its generalized numbers

“…In general for positive integer k, in paper [1], it is called as k-Fuss-Catalan numbers F n . The purpose of this paper is that we will give a determinant of normalized Hessenberg matrix in Section 2 for positive integer, and in Section 3 for rational k. In Section 4, we give the iterative method to evaluate the determinant.…”

“…2 The k-Fuss-Catalan numbers Given a positive number k, a k-Fuss-Catalan path of length n is a path from (0, 0) to (n, kn) using east steps (1, 0) and north steps (0, 1) such that it stays weakly below the line y = kx. The number of all k-Fuss-Catalan paths of length n is given by the k-Fuss-Catalan numbers [1],…”