Supercongruences involving Motzkin numbers and central trinomial coefficients

2022/08/22 by Ji-Cai Liu, Liu, Ji-Cai
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2208.10275

Abstract

Let M_n and T_n denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime p≥ 5, amp;∑_k=0^p-1M_k²≡ ((p)/(3))(2-6p)\pmodp²,
amp;∑_k=0^p-1kM_k²≡ ((p)/(3))(9p-1)\pmodp²,
amp;∑_k=0^p-1T_kM_k≡ (4)/(3)((p)/(3))+(p)/(6)(1-9((p)/(3)))\pmodp², where (-) is the Legendre symbol. These results confirm three 12-year-old supercongruence conjectures of Z.-W. Sun.

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