On the log-concavity of the sequence for some combinatorial sequences

Abstract

Recently, Sun posed a series of conjectures on the log-concavity of the sequence , where is a familiar combinatorial sequence of positive integers. Luca and Stănică, Hou et al. and Chen et al. proved some of Sun's conjectures. In this paper, we present a criterion on the log-concavity of the sequence . The criterion is based on the existence of a function f(n) that satisfies some inequalities involving terms related to the sequence . Furthermore, we present a heuristic approach to compute f(n). As applications, we prove that, for the Zagier numbers , the sequences are strictly log-concave, which confirms a conjecture of Sun. We also prove the log-concavity of the sequence of Cohen–Rhin numbers.