On the log-concavity of the sequence {n√Sn}n=1∞ for some combinatorial sequences {Sn}n=0∞

Recently, Sun posed a series of conjectures on the log-concavity of the sequence {(n)root S-n}(n=1)(infinity), where {S-n}(n=0)(infinity) is a familiar combinatorial sequence of positive integers. Luca and Stanica, 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 {(n)root S-n}(n=1)(infinity). The criterion is based on the existence of a function f(n) that satisfies some inequalities involving terms related to the sequence {S-n}(n=0)(infinity). Furthermore, we present a heuristic approach to compute f(n). As applications, we prove that, for the Zagier numbers {Z(n)}(n=0)(infinity), the sequences {n(root)Z(n)}(n=1)(infinity) are strictly log-concave, which confirms a conjecture of Sun. We also prove the log-concavity of the sequence {(n)root U-n}(n=1)(infinity) of Cohen-Rhin numbers.