On the log-concavity of the n-th root of sequences

Abstract

In recent years, the log-concavity of the n-th root of a sequence {Snn}n≥1 has been received a lot of attention. Recently, Sun posed the following conjecture in his new book: the sequences {ann}n≥2 and {bnn}n≥1 are log-concave, wherean:=1n∑k=0n−1(n−1k)2(n+kk)24k2−1 andbn:=1n3∑k=0n−1(3k2+3k+1)(n−1k)2(n+kk)2. In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a criterion on the log-concavity of {Snn}n≥1 given by us and a mathematica package due to Hou and Zhang, while the analytic method relies on a result due to Xia.