Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$

Nadav Kohen

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arXiv · 2403.00149

Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$

Nadav Kohen

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Abstract

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ 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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Nadav Kohen. 2024-02-29. Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$. https://arxiv.org/abs/2403.00149

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