Journal of Integer Sequences, Vol. 22 (2019), Article 19.5.8

Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles

Paul Barry
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.

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(Concerned with sequences A000108 A000984 A001006 A006318 A007318 A008288 A009766 A033184 A033282 A039598 A047891 A054726 A060693 A064063 A064641 A078740 A080247 A082298 A082301 A082302 A086810 A088617 A090442 A090452 A103210 A103211 A108524 A126216 A131198 A133305 A152600 A152601 A156017 A269730 A269731 A281260.)

Received December 20 2018; revised version received June 27 2019. Published in Journal of Integer Sequences, August 24 2019.

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