Motzkin numbers and related sequences modulo powers of $2$

TL;DR

This paper expresses the generating function of Motzkin numbers modulo powers of 2 as a polynomial in a basic series, enabling explicit computation of Motzkin numbers modulo 8 based on binary digits, and extends these results to related sequences.

Contribution

It provides a new polynomial expression for Motzkin numbers modulo powers of 2 and applies this to determine values modulo 8 using binary digits, extending previous results.

Findings

Motzkin numbers modulo 8 can be computed from binary digits of n.

The generating function can be expressed as a polynomial in a basic series.

Results extend to related combinatorial sequences.

Abstract

We show that the generating function $\sum_{n\ge0}M_n\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\sum _{e\ge0} ^{} {z^{4^e}}/( {1-z^{2\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Th\'eorie Nombres Bordeaux 27 (2015), 245-288]. Analogous results are also shown to hold for related combinatorial sequences, namely for the Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and for the sequence of hex tree numbers.