On a new congruence in the Catalan triangle

Jovan Mikić
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 667–682
DOI: 10.7546/nntdm.2025.31.3.667-682
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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

For 0\leq k \leq n, the number C(n,k) represents the number of all lattice paths in the plane from the point (0,0) to the point (n,k), using steps (1,0) and (0,1), that never rise above the main diagonal y=x. The Fuss–Catalan number of order three C^{(3)}_n represents the number of all lattice paths in the plane from the point (0,0) to the point (2n,n), using steps (1,0) and (0,1), that do not rise above the line y=\frac{x}{2}. The generalized Schröder number \mathrm{Schr}(n,m,2) of order two represents the number of all lattice paths in the plane from the point (0,0) to the point (n,m), using steps (1,0), (0,1), and (1,1), that never go below the line y=2x. We present a new alternating convolution formula for the numbers C(2n,k) multiplied by a power of a binomial coefficient. Using a new class of binomial sums that we call M sums, we prove that this sum is divisible by C^{(3)}_n and by the central binomial coefficient \binom{2n}{n}. We do this by examining the numbers T(n,j)=\frac{1}{2n+1}\binom{2n+j}{j}\binom{2n+1}{n+j+1}, for which we present a new combinatorial interpretation, connecting them to the generalized Schröder numbers of order two.

Keywords

  • Catalan triangle
  • Fuss–Catalan number of order three
  • Catalan number
  • Central binomial coefficient
  • M sum
  • Schröder number of order two
  • Lattice path
  • Induction principle

2020 Mathematics Subject Classification

  • 05A10
  • 11B65

References

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Manuscript history

  • Received: 4 March 2025
  • Revised: 28 September 2025
  • Accepted: 29 September 2025
  • Online First: 29 September 2025

Copyright information

Ⓒ 2025 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. (2025). On a new congruence in the Catalan triangle. Notes on Number Theory and Discrete Mathematics, 31(3), 667-682, DOI: 10.7546/nntdm.2025.31.3.667-682.

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