A Criterion for the Log-Convexity of Combinatorial Sequences

Abstract

Recently, Došlić, and Liu and Wang developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, in order to prove the log-convexity of the Apéry numbers $A_n$, by our method, we just need to evaluate the values of $A_n$ for $0\leq n \leq 6$. As applications, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by Sun.