Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.5

Dyck Paths, Motzkin Paths, and the Binomial Transform

Stefano Capparelli and Alberto Del Fra
Università di Roma La Sapienza


We study the moments of orthogonal polynomial sequences (OPS) arising from tridiagonal matrices. We obtain combinatorial information about the sequence of moments of some OPS in terms of Motzkin and Dyck paths, and also in terms of the binomial transform. We then introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of those equivalence classes, and use this information to obtain a combinatorial formula for the number of Dyck and Motzkin paths of a fixed length.

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(Concerned with sequences A025235 A025237 A063376 A151374.)

Received May 19 2015; revised versions received July 19 2015; July 27 2015. Published in Journal of Integer Sequences, July 29 2015.

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