On Motzkin numbers and central trinomial coefficients

TL;DR

This paper uncovers surprising arithmetic properties of Motzkin numbers and central trinomial coefficients, revealing new divisibility and integrality results with potential combinatorial implications.

Contribution

It establishes novel arithmetic identities and divisibility properties involving Motzkin numbers and central trinomial coefficients for all positive integers.

Findings

/n imes \u2211_{k=1}^n (2k+1) M_k^2 is an integer

/6 imes n^2(n^2-1) divides  ^{n-1} k(k+1)(8k+9) T_k T_{k+1}

^n-1 (k+1)(k+2)(2k+3) M_k 2^{n-1-k} equals n(n+1)(n+2) M_n M_{n-1}

Abstract

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z,$$ $$\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$