Factors of alternating convolution of the Gessel numbers

Jovan Mikić
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 857–868
DOI: 10.7546/nntdm.2024.30.4.857-868
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Authors and affiliations

Jovan Mikić
Faculty of Technology, Faculty of Natural Sciences and Mathematics, University of Banja Luka
Bosnia and Herzegovina

Abstract

The Gessel number P(n,r) is the number of lattice paths in the plane with (1,0) and (0,1) steps from (0,0) to (n+r, n+r-1) that never touch any of the points from the set \{(x,x) \in \mathbb{Z}^2: x \geq r\}. We show that there is a close relationship between Gessel numbers P(n,r) and super Catalan numbers T(n,r). A new class of binomial sums, so called M sums, is used. By using one form of the Pfaff–Saalschütz theorem, a new recurrence relation for M sums is proved. Finally, we prove that an alternating convolution of Gessel numbers P(n,r) multiplied by a power of a binomial coefficient is always divisible by \frac{1}{2}T(n,r).

Keywords

  • Gessel number
  • Super Catalan number
  • M sum
  • Catalan number
  • Pfaff–Saalschütz theorem

2020 Mathematics Subject Classification

  • 05A10
  • 11B65

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Manuscript history

  • Received: 3 July 2024
  • Revised: 3 December 2024
  • Accepted: 11 December 2024
  • Online First: 16 December 2024

Copyright information

Ⓒ 2024 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Mikić, J. (2024). Factors of alternating convolution of the Gessel numbers. Notes on Number Theory and Discrete Mathematics, 30(4), 857-868, DOI: 10.7546/nntdm.2024.30.4.857-868.

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