Hyper-Leonardo numbers: Combinatorial interpretation and some positivities

Abstract

In this paper, we establish the combinatorial interpretations of the hyper-Leonardo numbers $Le_n^{(r)}$ and Leonardo numbers $Le_n$. We investigate the log-concavity of the Leonardo numbers for $n\geq 3$ and the hyper-Leonardo numbers for $n,r\geq 1$. In addition, we prove the log-balancedness of the hyper-Leonardo numbers for $r=1,2$. Furthermore, we prove the $q$-log-concavity of the polynomial $\sum_{k=0}^{n}Le_k^{(r)} q^k$ for $n,r\geq 1$.