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 $ConstantTermOf\left[P(x)^nQ(x)\right]$ 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 $p$.