The Log-Behavior of the Catalan–larcombe–french Sequence

Abstract

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

Log-behavior of two sequences related to the elliptic integrals

“…The log-convexity of {P n } n≥0 , conjectured by Sun [14], has been proved by Xia and Yao [15] and independently by Zhao [18]. The log-concavity of {V n } n≥1 , conjectured by Zhao [19], has been confirmed by Yang and Zhao [17].…”

“…18 + 57n 17 + 96n 16 + 234n 15 + 706n 14 + 1908n 13 + 2616n 12 + 3126n 11 + 8130n 10 + 18198n 9 + 27248n 8 + 14970n 7 + 5478n 6 + 49572n 5 + 97308n 4 + 77760n 3 − 58752n 2 − 165888n − 82944.…”

Log-concavity of the Fennessey-Larcombe-French Sequence

“…For instance, Xia and Yao [13] obtained the log-convexity of the Catalan-Larcombe-French sequence, and confirmed a conjecture of Sun [11]. By using a log-balancedness criterion due to Došlić [4], Zhao [16] proved the log-balancedness of the Catalan-Larcombe-French sequence.…”

Sun’s log-concavity conjecture on the Catalan–Larcombe–French sequence

“…The strictly increasing property of the sequence {P n+1 /P n } n≥0 and { n √ P n } n≥1 has been confirmed by Xia and Yao [12], and independently by Zhao [14]. We would like to point out that Zhao [14] only proved the log-convexity of {P n } n≥0 , which also implies the monotonicity of { n √ P n } n≥1 by using a result due to Wang and Zhu [11].…”