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 in the form ConstantTermOf (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.