Uniform Recurrence in the Motzkin Numbers and Related Sequences mod \(p\)

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 \(p\).

Published here.