On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers

Yang Wen

doi:doi:10.11648/j.ajmcm.20170204.11

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On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers

Yang Wen

Received: 29 October 2016 Accepted: 31 March 2017 Published: 17 April 2017

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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.

DOI	10.11648/j.ajmcm.20170204.11
Published in	American Journal of Mathematical and Computer Modelling (Volume 2, Issue 4, November 2017)
Page(s)	95-98
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

The Catalan-Larcombe-French Number, Log-Concavity, Recurrence Relation

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.

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APA Style

Yang Wen. (2017). On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers. American Journal of Mathematical and Computer Modelling, 2(4), 95-98. https://doi.org/10.11648/j.ajmcm.20170204.11

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ACS Style

Yang Wen. On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers. Am. J. Math. Comput. Model. 2017, 2(4), 95-98. doi: 10.11648/j.ajmcm.20170204.11

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AMA Style

Yang Wen. On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers. Am J Math Comput Model. 2017;2(4):95-98. doi: 10.11648/j.ajmcm.20170204.11

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@article{10.11648/j.ajmcm.20170204.11,
  author = {Yang Wen},
  title = {On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {4},
  pages = {95-98},
  doi = {10.11648/j.ajmcm.20170204.11},
  url = {https://doi.org/10.11648/j.ajmcm.20170204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170204.11},
  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.},
 year = {2017}
}

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TY  - JOUR
T1  - On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers
AU  - Yang Wen
Y1  - 2017/04/17
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170204.11
DO  - 10.11648/j.ajmcm.20170204.11
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
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PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170204.11
AB  - 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.
VL  - 2
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Author Information

Yang Wen

Department of Automobile Engineering and Transport, Lanzhou Vocational Technical College, Lanzhou, P. R. China

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Plain Text BibTeX RIS

APA Style

Yang Wen. (2017). On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers. American Journal of Mathematical and Computer Modelling, 2(4), 95-98. https://doi.org/10.11648/j.ajmcm.20170204.11

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ACS Style

Yang Wen. On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers. Am. J. Math. Comput. Model. 2017, 2(4), 95-98. doi: 10.11648/j.ajmcm.20170204.11

Copy | Download

AMA Style

Yang Wen. On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers. Am J Math Comput Model. 2017;2(4):95-98. doi: 10.11648/j.ajmcm.20170204.11

Copy | Download

@article{10.11648/j.ajmcm.20170204.11,
  author = {Yang Wen},
  title = {On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {4},
  pages = {95-98},
  doi = {10.11648/j.ajmcm.20170204.11},
  url = {https://doi.org/10.11648/j.ajmcm.20170204.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170204.11},
  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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - On the Log-Concavity of the Root of the Catalan-Larcombe-French Numbers
AU  - Yang Wen
Y1  - 2017/04/17
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170204.11
DO  - 10.11648/j.ajmcm.20170204.11
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 95
EP  - 98
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170204.11
AB  - 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.
VL  - 2
IS  - 4
ER  -

Copy | Download