Density and Symmetry in the Generalized Motzkin Numbers mod $p$

We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a, b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a, b}_n\bmod p$ (with $n<p$). We apply our method to various other sequences to obtain similar formulas. We also prove that $T^{a, b}_{p-1-n}\equiv (b^2-4a^2)^{\frac{p-1}{2}-n}T^{a, b}_n\pmod p$ to obtain tight lower bounds for the density of $0$ in our sequences. This symmetry of the first $p$ central trinomial coefficients mod $p$ also appears in a couple of other applications, including the proof of a novel symmetry of the first $p-2$ Motzkin numbers that is of independent interest: $M^{a, b}_{p-3-n}\equiv (b^2-4a^2)^{\frac{p-3}{2}-n}M^{a, b}_n\pmod p$.