Uniform Recurrence in the Motzkin Numbers and Related Sequences mod p

2024/02/29 by Nadav Kohen, Kohen, Nadav
Mathematics · #05A15 #11B50 (Primary) 68R15 #11B85 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2403.00149

openalex publication_date 2024/02/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form ConstantTermOf[P(x)ⁿQ(x)] for Laurent polynomials Q, and symmetric Laurent trinomials P. In this paper we characterize the primes for which sequences of this form are uniformly recurrent modulo p. For all other primes, we show that 0 has density 1. This will be accomplished by showing that the study of these sequences mod p can be reduced to the study of the generalized central trinomial coefficients, which are well-behaved mod p.

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