Density and Symmetry in the Generalized Motzkin Numbers mod $p$
Nadav Kohen
TL;DR
This work addresses the problem of determining how often the generalized Motzkin numbers $M^{a,b}_n$ are divisible by a prime $p$ by reducing the analysis to the first $p$ values of the generalized central trinomial coefficients $T^{a,b}_n$ via the identity $2a^2M^{a,b}_n = (4a^2-b^2)T^{a,b}_n + 2bT^{a,b}_{n+1} - T^{a,b}_{n+2}$. A key symmetry is established: $T^{a,b}_{p-1-k} \equiv (b^2-4a^2)^{\frac{p-1}{2}-k}T^{a,b}_k \pmod p$ for $p>2$, along with a corresponding symmetry for $M^{a,b}_{p-3-k}$, enabling a Lucas-type reduction that ties zero-density in $M^{a,b}_n\bmod p$ to the first $p$ central trinomial coefficients. The paper then provides a concrete formula for the density of zeros, $D_0$, in terms of these first $p$ coefficients, with a dichotomy: if some $T^{a,b}_n\equiv 0$, the density is $1$, otherwise $D_0$ is given by explicit combinatorial counts; under a generation assumption for $T^{a,b}_n$, nonzero residues attain density $(1-D_0)/(p-1)$, and a universal lower bound $D_0 \ge 2/(p(p-1))$ holds. The methodology extends to other sequences (e.g., Riordan numbers and related OEIS sequences), demonstrating the broad applicability and revealing equal densities across certain pairs of sequences, thereby offering a general toolkit for modular density questions in combinatorial sequences.
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: $M^{a, b}_{p-3-n}\equiv (b^2-4a^2)^{\frac{p-3}{2}-n}M^{a, b}_n\pmod p$.