Infinitely log-monotonic combinatorial sequences

Abstract

Abstract. We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {a n } n≥0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {a n+1 /a n } n≥0 is log-concave. Furthermore, we prove that if a sequence {a n } n≥k is ratio log-concave, then the sequence { n √ a n } n≥k is strictly logconcave subject to a certain initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers D n , we confirm a conjecture of Sun on the log-concavity of the sequence { n √ D n } n≥1 .

The log-behavior of $$\root n \of {p(n)}$$ p ( n ) n and $$\root n \of {p(n)/n}$$

“…For n ≥ 50, the upper bound in (1.5) can be relaxed to 24π (24n) 3 By using the Lambert W function, it can be shown that − 1 n 2 + 3 n 5/2 + 2e − π 10 √ 2n 3 < 0 when n ≥ 5000, and therefore we arrive at the upper bound in the form of (1.4). Let r(n) = n p(n)/n.…”

“…It should be noted that there is another approach to proving the log-convexity of { n p(n)} n≥27 . Chen, Guo and Wang [3] introduced the notion of a ratio log-convex sequence and showed that ratio log-convexity implies log-convexity under an initial condition. A sequence {a n } n≥k is called ratio log-convex if {a n+1 /a n } n≥k is log-convex, or, equivalently, for n ≥ k, log a n+2 − 3 log a n+1 + 3 log a n − log a n−1 ≥ 0.…”

Sun’s log-concavity conjecture on the Catalan–Larcombe–French sequence

“…The main objective of this paper is to prove the log-concavity of the sequence { n √ P n } n≥1 and the sequence { n √ V n } n≥1 , where {P n } n≥0 and {V n } n≥0 , known as the Catalan-Larcombe-French sequence and the Fennessey-Larcombe-French sequence respectively, are given by (n + 1) 2 P n+1 = 8(3n 2 + 3n + 1)P n − 128n 2 P n−1 , (1.1) n(n + 1) 2 V n+1 = 8n(3n 2 + 5n + 1)V n − 128(n − 1)(n + 1) 2…”

“…For a positive sequence {a n } n≥0 satisfying a three-term recurrence relation, Chen, Guo and Wang [2] obtained a useful criterion to determine the log-concavity of { n √ a n } n≥N for some positive integer N. While, their criterion does not apply to the the Catalan-Larcombe-French sequence, and Sun's conjecture on the log-concavity of { n √ P n } n≥1 remains open. The first main result of this paper is as follows.…”

Log-concavity of the Fennessey-Larcombe-French Sequence

“…We first give a sufficient condition for log-concavity of a positive sequence subject to certain three-term recurrence. It should be mentioned that the log-behavior of sequences satisfying three-term recurrences has been extensively studied, see Liu and Wang [10], Chen and Xia [3], Chen, Guo and Wang [2], and Wang and Zhu [12]. However, most of these studies have focused on the log-convexity of such sequences instead of their log-concavity.…”