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