Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T21:15:03.288Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)