Volume 2, Issue 4, November 2017, Page: 95-98
On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers
Yang Wen, Department of Automobile Engineering and Transport, Lanzhou Vocational Technical College, Lanzhou, P. R. China
Received: Oct. 29, 2016;       Accepted: Mar. 31, 2017;       Published: Apr. 17, 2017
DOI: 10.11648/j.ajmcm.20170204.11      View  1616      Downloads  121
Abstract
Recently, some combinatorial properties for the the Catalan-Larcombe-French numbers have been proved by Sun and Wu, and Zhao. Recently, Z. W. Sun conjectured that the root of the Catalan-Larcombe-French numbers is log-concave. In this paper, we confirm Sun's conjecture by establishing the lower and upper bound for the ratios of the Catalan-Larcombe-French numbers.
Keywords
The Catalan-Larcombe-French Number, Log-Concavity, Recurrence Relation
To cite this article
Yang Wen, On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers, American Journal of Mathematical and Computer Modelling. Vol. 2, No. 4, 2017, pp. 95-98. doi: 10.11648/j.ajmcm.20170204.11
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between tworecent integer sequences, Congr. Numer. 169 (2004) 79-99.
[2]
P. Larcombe and D. R. French, On the `other' Catalan numbers: a historicalformulation re-examined, Congr. Numer. 143 (2000) 33-64.
[3]
P. Larcombe and D. R. French, On the integrality of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816,…}, Cong. Num. 148 (2001) 65-91.
[4]
P. Larcombe and D. R. French, A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic, Cong. Num. 166 (2004)161-172.
[5]
P. Larcombe, D. R. French and E. J. Fennessey, The asymptotic behaviour of the Catalan-Larcombe-French sequence {1, 8, 80, 896, 10816,….}, Util. Math. 60 (2001) 67-77.
[6]
P. Larcombe, D. R. French and C. A. Woodham, A note on the asymptotic behaviour of a prime factor decomposition of the general Catalan-Larcombe-Frenchnumber, Cong. Num. 156 (2002) 17-25.
[7]
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at www.research.att.com/vjas/sequences/.
[8]
B. Y. Sun and B. Wu, Two-log-convexity of the Catalan-Larcombe-French sequence, J. Ineq. Appl. 2015 (2015) # P404.
[9]
M. R. Sun and L. J. Jin, Proof of a conjecture on the Catalan-Larcombe-Frenchnumbers, Ars Combin., to appear.
[10]
Z. W. Sun, Conjectures involving arithmetical sequences, Numbers Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H. Li and J. Liu), Proc. 6thChina-Japan Seminar (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258.
[11]
E. X. W. Xia and O. X. M. Yao, A criterion for the log-convexity of combinatorial sequences, Electr. J. Combin. 20 (4) (2014) # P3.
[12]
F. Z. Zhao, The log-behavior of the Catalan-Larcombe-French sequences, Int. J. Number Theory 10 (2014) 177-182.
Browse journals by subject