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

Abstract

Many famous integer sequences, including the Catalan numbers and the Motzkin numbers, can be expressed as the constant terms of the polynomials $P(x)^nQ(x)$ for some Laurent polynomial $Q$, and symmetric Laurent trinomial $P$. 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 the set of indices for which our sequences are congruent to $0$ has density $1$. This is 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 $p$.