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

Part of: Combinatorics

Published online by Cambridge University Press:  22 June 2018

Ernest X. W. Xia
Ernest X. W. Xia*
Affiliation:
Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, People's Republic of China ([email protected])
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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.

Keywords

combinatorial sequencesmonotonicitylog-concavitylog-convexity

MSC classification

Secondary: 05A20: Combinatorial inequalities 05A10: Factorials, binomial coefficients, combinatorial functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 4 , August 2018 , pp. 881 - 892
Copyright
Copyright © Royal Society of Edinburgh 2018 

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