Proof of a conjecture of Z.W. Sun

Abstract

Rec ently, Sun defined a newsequence $a(n)= \sum_{k=0}^n {n\choose 2k}{2k\choose k}\frac{1}{2k-1} $, which can be viewed as an analogue of Motzkin numbers. Sun conjectured that the sequence $\{\frac{a(n+1)}{a(n)}\}_{n\geq 5} $ is strictly increasing with limit 3, and the sequence $\{ \sqrt[n+1]{a(n+1)}/\sqrt[n]{a(n)}\}_{n\geq 9} $ is strictly decreasing with limit 1. In this paper, we confirm Sun's conjecture.