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Supercongruences involving Motzkin numbers and central trinomial coefficients

Part of: Combinatorics Sequences and sets Elementary number theory

Published online by Cambridge University Press:  03 October 2024

Ji-Cai Liu
Ji-Cai Liu*
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, 325035, PR China
*
Email: jcliu2016@gmail.com
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Abstract

Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime $p\ge 5$,

\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}

where $\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.

Keywords

Motzkin numbersCatalan numberscentral trinomial coefficientssupercongruencesLegendre symbol

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 05A19: Combinatorial identities, bijective combinatorics 11B65: Binomial coefficients; factorials; $q$-identities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 4 , November 2024 , pp. 1060 - 1084
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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