Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles
School of Science
Waterford Institute of Technology
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.
Full version: pdf,
(Concerned with sequences
Received December 20 2018; revised version received June 27 2019.
Published in Journal of Integer Sequences, August 24 2019.
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