Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.4

When Numerical Analysis Crosses Paths with Catalan and Generalized Motzkin Numbers

Paul Eloe and Catherine Kublik
Department of Mathematics
University of Dayton
300 College Park
Dayton, OH 45469


We study a linear doubly indexed sequence that contains the Catalan numbers and relates to a class of generalized Motzkin numbers. We obtain a closed form formula, a generating function and a nonlinear recursion relation for this sequence. We show that a finite difference scheme with compact stencil applied to a nonlinear differential operator acting on the Euclidean distance function is exact, and exploit this exactness to produce the nonlinear recursion relation. In particular, the nonlinear recurrence relation is obtained by using standard error analysis techniques from numerical analysis. This work shows a connection between numerical analysis and number theory, and illustrates an interesting occurrence of the Catalan and generalized Motzkin numbers in a context a priori void of combinatorial objects.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000108 A001006.)

Received May 11 2018; revised version received August 27 2018. Published in Journal of Integer Sequences, November 24 2018.

Return to Journal of Integer Sequences home page