Seven (Lattice) Paths to Log-Convexity

Abstract

Three new methods for proving log-convexity of combinatorial sequences are presented. Their implementation is demonstrated and their performance is compared with four more familiar approaches in the context of sequences that enumerate various classes of lattice paths.

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

“…The nth trinomial coefficient T n is the coefficient of x n in the expansion (1 + x + x 2 ) n . It is known that (n + 1)T n+1 = (2n + 1)T n + 3nT n−1 (2 We refer the reader to [7] for another proof of the log-convexity of {T n } n≥4 .…”

A Criterion for the Log-Convexity of Combinatorial Sequences

“…In 1994, Engel [8] proved the log-convexity of the Bell numbers. Recently, Došlić [4,5,6], Došlić and Veljan [7], and Liu and Wang [14] developed techniques for proving the log-convexity of sequences. Došlić [4,5,6] presented several methods for dealing with the log-convexity of combinatorial sequences.…”

“…Recently, Došlić [4,5,6], Došlić and Veljan [7], and Liu and Wang [14] developed techniques for proving the log-convexity of sequences. Došlić [4,5,6] presented several methods for dealing with the log-convexity of combinatorial sequences. He proved that the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4, the Apéry numbers, the large Schröder numbers, the derangements numbers and the central Delannoy numbers are log-convex.…”

“…and k, l, a k , b k , c l and d l are positive numbers. The authors [19] gave a criterion for the positivity of the sequence {S n } ∞ n=0 defined by (6). The aim of this paper is to present a criterion for the logconvexity of some famous combinatorial sequences.…”

“…The aim of this paper is to present a criterion for the logconvexity of some famous combinatorial sequences. By our method, in order to determine the log-convexity of the sequence {S n } ∞ n=0 defined by (6), it suffices to compute a constant number of terms at the beginning of the sequence {S n } ∞ n=0 . As applications, we prove some famous combinatorial sequences are strictly log-convex.…”