The 2-log-convexity of the Apery Numbers

TL;DR

This paper introduces a method to prove 2-log-convexity for sequences with three-term recurrence relations and demonstrates its effectiveness on several well-known combinatorial sequences.

Contribution

It provides a new approach to establish 2-log-convexity for important sequences, extending the understanding of their convexity properties.

Findings

Apery numbers are 2-log-convex.

Cohen-Rhin, Motzkin, Fine, Franel (order 3 and 4), and large Schroder numbers are 2-log-convex.

Numerical evidence indicates these sequences are k-log-convex for all k ≥ 1, except possibly for initial terms.

Abstract

We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schroder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are k-log-convex for any $k\geq 1$ possibly except for a constant number of terms at the beginning.