Proof of a conjecture of Z.W. Sun
DOI:
https://doi.org/10.11575/cdm.v14i1.62700Abstract
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.
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