Infinitely log-monotonic combinatorial sequences

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 {an}n⩾0{an}n⩾0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {an+1/an}n⩾0{an+1/an}n⩾0 is log-concave. Furthermore, we prove that if a sequence {an}n⩾k{an}n⩾k is ratio log-concave, then the sequence {ann}n⩾k is strictly log-concave subject to an 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 DnDn, we confirm a conjecture of Sun on the log-concavity of the sequence {Dnn}n⩾1.