Sun's log-concavity conjecture on the Catalan-Larcombe-French sequence (Q287964)

In the paper under review, the author proved a conjecture of \textit{Z.-W. Sun} [in: Number theory. Arithmetic in Shangri-La. Proceedings of the 6th China-Japan seminar held at Shanghai Jiao Tong University, Shanghai, China, August 15--17, 2011. Hackensack, NJ: World Scientific. 244--258 (2013; Zbl 1308.11031)], which states that the sequence \(\sqrt[n]{P_n}_{n\geq 1}\) is strictly log-concave, where \(P_n\) denotes the \(n\)-th Catalan-Larcombe-French number. Remarkably, the author proved that \[ \frac{16(n-2)}{n-1}<\frac{P_n}{P_{n-1}}<\frac{16(n-1)}{n}, \] from which the strict log-concavity of \(\sqrt[n]{P_n}_{n\geq 1}\) follows. The proof is very elegant, and this paper is a nice contribution to the theory of Catalan-Larcombe-French numbers. Similar results for the Fennessey-Larcombe-French sequences are also obtained.