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THE LOG-BEHAVIOR OF THE CATALAN–LARCOMBE–FRENCH SEQUENCE

FENG-ZHEN ZHAO

Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China

https://doi.org/10.1142/S1793042113500887Cited by:5 (Source: Crossref)

Abstract

In this paper, we discuss the log-behavior of the Catalan–Larcombe–French sequence {P_n}_n≥0. We prove that {P_n}_n≥0 is log-balanced and {P_n/(n!)²}_n≥0 is unimodal. In addition, we show that {P_k/k!}_0≤k≤n is reverse ultra log-concave.

Keywords:

AMSC: 05A20, 11B37, 11B83

References

N. Asai, I. Kubo and H. H. Kubo, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4(1), 59 (2001), DOI: 10.1142/S0219025701000498. Link, Web of Science, Google Scholar
E. Catalan, Rend. Circ. Mat. Palermo 1(1), 190 (1887), DOI: 10.1007/BF03020089. Crossref, Google Scholar
W. Y. C. Chen and C. C. Y. Gu, Proc. Amer. Math. Soc. 137(12), 3991 (2009), DOI: 10.1090/S0002-9939-09-09976-6. Crossref, Web of Science, Google Scholar
T. Došlić, Int. J. Math. Math. Sci. 4, 507 (2005). Google Scholar
T. Došlić and D. Veljan, Discrete Math. 308(11), 2182 (2008), DOI: 10.1016/j.disc.2007.04.066. Crossref, Web of Science, Google Scholar
H. Han and S. Seo, European J. Combin. 29(7), 1544 (2008), DOI: 10.1016/j.ejc.2007.12.002. Crossref, Web of Science, Google Scholar
A. F. Jarvis, P. Larcombe and D. French, Congr. Numer. 161, 151 (2003). Google Scholar
A. F. Jarvis, P. Larcombe and D. French, Congr. Numer. 169, 79 (2004). Google Scholar
A. F. Jarvis, P. Larcombe and D. French, Indian J. Math. 47(3), 159 (2005). Google Scholar
F. Jarvis and H. A. Verrill, Ramanujan J. 22(2), 171 (2010), DOI: 10.1007/s11139-009-9218-5. Crossref, Web of Science, Google Scholar
J. R. Klauder, A. Penson and J. M. Sixdeniers, Phys. Rev. A 64(1), 013817 (2001), DOI: 10.1103/PhysRevA.64.013817. Crossref, Web of Science, Google Scholar
H. H. Kuo , White Noise Distribution Theory ( CRC Press , Boca Raton, FL , 1996 ) . Google Scholar
C. Lane, J. Integer Seq. 7(2), 5 (2004). Web of Science, Google Scholar
P. Larcombe and D. French, Congr. Numer. 143, 33 (2000). Google Scholar
K. A. Penson and A. I. Solomon, Coherent states from combinatorial sequences, Proc. 2nd Int. Symp. Quantum Theory and Symmetries, eds. E. Kapuscik and A. Horzela (World Scientific, Singapore, 2002) pp. 527–530. Google Scholar
Z. W. Sun, Conjectures involving arithmetical sequences, Proc. 6th China–Japan Seminar, eds. S. Kanemitsu, H. Li and J. Liu (World Scientific, Singapore, 2013) pp. 244–258. Google Scholar