The Log-Behavior of Some Sequences Related to The Catalan-Larcombe-French Sequence and the Fennessey-Larcombe-French Sequence

Abstract

In this paper, for the Catalan-Larcombe-French sequence $\{P_n{\}}_{n\ge 0}$ and the Fennessey-Larcombe-French sequence $\{V_n{\}}_{n\ge 0}$, we mainly discuss the log-behavior of some sequences related to $\{P_n{\}}_{n\ge 0}$ and $\{V_n{\}}_{n\ge 0}$. For example, we study the log-behavior of some sequences such as $\{P_n^2{\}}_{n\ge 0}$, $\{n!n{V}_n{\}}_{n\ge 1}$, $\{n!V_n{\}}_{n\ge 0}$, and $\{V_n-P_n{\}}_{n\ge 2}$. In addition, we discuss the monotonicity of some sequences involving $\{P_n{\}}_{n\ge 0}$ and $\{V_n{\}}_{n\ge 0}$.