Please enter verification code
Confirm
On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers
American Journal of Mathematical and Computer Modelling
Volume 2, Issue 4, November 2017, Pages: 95-98
Received: Oct. 29, 2016; Accepted: Mar. 31, 2017; Published: Apr. 17, 2017
Views 2616      Downloads 193
Author
Yang Wen, Department of Automobile Engineering and Transport, Lanzhou Vocational Technical College, Lanzhou, P. R. China
Article Tools
Follow on us
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.
References
[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.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186