23 11 Article 19.5.8 Journal of Integer Sequences, Vol. 22 (2019), 2 3 6 1 47 Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles Paul Barry School of Science Waterford Institute of Technology Ireland pbarry@wit.ie Abstract We find closed-form expressions and continued fraction generating functions for a family of generalized Catalan numbers associated with a set of Pascal-like number triangles that are defined by Riordan arrays. We express these generalized Catalan numbers as the moments of appropriately defined orthogonal polynomials. We also describe them as the row sums of related Riordan arrays. Links are drawn to the Narayana numbers and to lattice paths. We further generalize this one-parameter family to a three-parameter family. We use the generalized Catalan numbers to define generalized Catalan triangles. We define various generalized Motzkin numbers defined by these general Catalan numbers. Finally we indicate that the generalized Catalan numbers can be associated with certain generalized Eulerian numbers by means of a special transform. 1 Introduction The Catalan numbers [26] are among the most important numbers in combinatorics. They have many important properties, which are shared to one degree or another with related sequences. This has prompted works which extend or generalize the Catalan numbers in various ways [1, 13]. In this note, we use the theory of Riordan arrays, and in particular a family of Pascal-like Riordan arrays, to find families of generalized Catalan numbers. A short introduction to Riordan arrays is provided later in this section. In fact, we find families of Catalan-like polynomials, which through specialization, give us generalized Catalan numbers. All are found to be moment sequences, like the Catalan 1