Proof of a Conjecture on the Catalan-Larcombe-French Numbers

Abstract

Let \(P_n\) denote the \(n\)-th Catalan-Larcombe-French number. Recently, the \(2\)-log-convexity of the Catalan-Larcombe-French sequence was proved by Sun and Wu. Moreover, they also conjectured that the quotient sequence \(\{\frac{P_{n}}{P_{n-1}}\}_{n= 0}^\infty\) of the Catalan-Larcombe-French sequence is log-concave. In this paper, this conjecture is confirmed by utilizing the upper and lower bounds for \(\frac{P_{n}}{P_{n-1}}\) and finding a middle function \(f(n)\).