We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a sequence subject to certain three-term recurrence.
References
M. E. Catalan, Sur les nombres de segner, Rend. Circ. Mat. Palermo 1 (1887), no. 1, 190–201. 10.1007/bf03020089 19.0170.03 M. E. Catalan, Sur les nombres de segner, Rend. Circ. Mat. Palermo 1 (1887), no. 1, 190–201. 10.1007/bf03020089 19.0170.03
W. Y. C. Chen, J. J. F. Guo and L. X. W. Wang, Infinitely log-monotonic combinatorial sequences, Adv. in Appl. Math. 52 (2014), 99–120. 10.1016/j.aam.2013.08.003 MR3131928 1281.05019 W. Y. C. Chen, J. J. F. Guo and L. X. W. Wang, Infinitely log-monotonic combinatorial sequences, Adv. in Appl. Math. 52 (2014), 99–120. 10.1016/j.aam.2013.08.003 MR3131928 1281.05019
W. Y. C. Chen and E. X. W. Xia, The $2$-log-convexity of the Apéry numbers, Proc. Amer. Math. Soc. 139 (2011), no. 2, 391–400. 10.1090/S0002-9939-2010-10575-0 MR2736323 W. Y. C. Chen and E. X. W. Xia, The $2$-log-convexity of the Apéry numbers, Proc. Amer. Math. Soc. 139 (2011), no. 2, 391–400. 10.1090/S0002-9939-2010-10575-0 MR2736323
T. Došlić, Log-balanced combinatorial sequences, Int. J. Math. Math. Sci. 2005 (2005), no. 4, 507–522. 10.1155/IJMMS.2005.507 T. Došlić, Log-balanced combinatorial sequences, Int. J. Math. Math. Sci. 2005 (2005), no. 4, 507–522. 10.1155/IJMMS.2005.507
A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congr. Numer. 169 (2004), 79–99. MR2122061 1070.11009 A. F. Jarvis, P. J. Larcombe and D. R. French, Linear recurrences between two recent integer sequences, Congr. Numer. 169 (2004), 79–99. MR2122061 1070.11009
F. Jarvis and H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J. 22 (2010), no. 2, 171–186. 10.1007/s11139-009-9218-5 1218.11025 F. Jarvis and H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J. 22 (2010), no. 2, 171–186. 10.1007/s11139-009-9218-5 1218.11025
P. J. Larcombe, D. R. French and E. J. Fennessey, The Fennessey-Larcombe-French sequence $\set{1, 8, 144, 2432, 40000, \ldots}$: Formulation and asymptotic form, Congr. Numer. 158 (2002), 179–190. MR1985157 1064.11021 P. J. Larcombe, D. R. French and E. J. Fennessey, The Fennessey-Larcombe-French sequence $\set{1, 8, 144, 2432, 40000, \ldots}$: Formulation and asymptotic form, Congr. Numer. 158 (2002), 179–190. MR1985157 1064.11021
––––, The Fennessey-Larcombe-French sequence $\set{1, 8, 144, 2432, 40000, \ldots}$: A recursive formulation and prime factor decomposition, Congr. Numer. 160 (2003), 129–137. MR2049108 1047.11020 ––––, The Fennessey-Larcombe-French sequence $\set{1, 8, 144, 2432, 40000, \ldots}$: A recursive formulation and prime factor decomposition, Congr. Numer. 160 (2003), 129–137. MR2049108 1047.11020
L. L. Liu and Y. Wang, On the log-convexity of combinatorial sequences, Adv. in Appl. Math. 39 (2007), no. 4, 453–476. 10.1016/j.aam.2006.11.002 1131.05010 L. L. Liu and Y. Wang, On the log-convexity of combinatorial sequences, Adv. in Appl. Math. 39 (2007), no. 4, 453–476. 10.1016/j.aam.2006.11.002 1131.05010
R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph Theory and its Applications: East and West (Jinan, 1986), 500–535, Ann. New York Acad. Sci. 576, New York Acad. Sci., New York, 1989. 10.1111/j.1749-6632.1989.tb16434.x 0792.05008 R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph Theory and its Applications: East and West (Jinan, 1986), 500–535, Ann. New York Acad. Sci. 576, New York Acad. Sci., New York, 1989. 10.1111/j.1749-6632.1989.tb16434.x 0792.05008
Z.-W. Sun, Conjectures involving arithmetical sequences, in Numbers Theory: Arithmetic in Shangri-La, 244–258, Ser. Number Theory Appl. 8, World Sci., Hackensack, NJ, 2013. 10.1142/9789814452458_0014 Z.-W. Sun, Conjectures involving arithmetical sequences, in Numbers Theory: Arithmetic in Shangri-La, 244–258, Ser. Number Theory Appl. 8, World Sci., Hackensack, NJ, 2013. 10.1142/9789814452458_0014
Y. Wang and B. Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, Sci. China Math. 57 (2014), no. 11, 2429–2435. 10.1007/s11425-014-4851-x MR3266503 1306.05015 Y. Wang and B. Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, Sci. China Math. 57 (2014), no. 11, 2429–2435. 10.1007/s11425-014-4851-x MR3266503 1306.05015
E. X. W. Xia and O. X. M. Yao, A criterion for the log-convexity of combinatorial sequences, Electron. J. Combin. 20 (2013), no. 4, Paper 3, 10 pp. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p3 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p3 MR3139388 1298.05041 E. X. W. Xia and O. X. M. Yao, A criterion for the log-convexity of combinatorial sequences, Electron. J. Combin. 20 (2013), no. 4, Paper 3, 10 pp. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p3 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p3 MR3139388 1298.05041
D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and Symmetries, 349–366, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009. 1244.11042 D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and Symmetries, 349–366, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009. 1244.11042
F.-Z. Zhao, The log-behavior of the Catalan-Larcombe-French sequences, Int. J. Number Theory 10 (2014), no. 1, 177–182. 10.1142/S1793042113500887 MR3189974 1329.11012 F.-Z. Zhao, The log-behavior of the Catalan-Larcombe-French sequences, Int. J. Number Theory 10 (2014), no. 1, 177–182. 10.1142/S1793042113500887 MR3189974 1329.11012
––––, The log-balancedness of combinatorial sequences, The 6th National Conference on Combinatorics and Graph Theory, Guangzhou, November 7–10, 2014, China. ––––, The log-balancedness of combinatorial sequences, The 6th National Conference on Combinatorics and Graph Theory, Guangzhou, November 7–10, 2014, China.
