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

TL;DR

This paper derives formulas for the density of zeroes in generalized Motzkin numbers modulo a prime, revealing symmetries in these sequences and related combinatorial numbers.

Contribution

It introduces a new formula linking the density of zeroes in generalized Motzkin numbers to central trinomial coefficients modulo a prime, and uncovers novel symmetry properties.

Findings

Density of zeroes in sequences is expressed via initial coefficients.

Symmetry relations for central trinomial and Motzkin numbers modulo p.

Lower bounds for zero density based on symmetry properties.

Abstract

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: $M^{a, b}_{p-3-n}\equiv (b^2-4a^2)^{\frac{p-3}{2}-n}M^{a, b}_n\pmod p$.