Logarithmic behavior of some combinatorial sequences

Abstract

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus.Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schröder numbers, sequences of values of some classic orthogonal polynomials, and many others. The calculus method extends also to two-(or more-) indexed sequences.

The 2-log-convexity of the Apéry Numbers

“…The proof is similar to that of Theorem 5.1, and it is omitted. Došlić [7,8] has proved the log-convexity of several well-known sequences of combinatorial numbers such as the Motzkin numbers M n , the Fine numbers F n , the Franel numbers F (3) n and F (4) n of order 3 and 4, and the large Schröder numbers s n . Based on the recurrence relations satisfied by these numbers, we utilize Theorem 2.1 to deduce that these sequences are all strictly 2-log-convex possibly except for a fixed number of terms at the beginning.…”

Seven (Lattice) Paths to Log-Convexity

“…As is well known [47], the class of P -recursive sequences is very rich and contains many combinatorially relevant sequences. Also, it was shown in [22] how to extend the calculus method to work for two-indexed sequences, such as the triangle of Eulerian numbers. Finally, at the price of lengthening the recursion, the methods can be applied also on non-homogeneous recurrences [22].…”

“…Also, it was shown in [22] how to extend the calculus method to work for two-indexed sequences, such as the triangle of Eulerian numbers. Finally, at the price of lengthening the recursion, the methods can be applied also on non-homogeneous recurrences [22]. Hence, all three methods are widely applicable.…”

“…This, in turn, means that the sequence x n is increasing, and that the sequence W n is log-convex. The interlacing method has been successfully applied to establish the log-convexity of sequences of secondary structures [23], sequences counting directed column-convex polyominos [20], derangements, and certain classes of symmetric matrices with integer entries [22].…”

Asymptotic r-log-convexity and P-recursive sequences

“…Recall that a P-recursive sequence of order d satisfies a recurrence relation of the form a n = r 1 (n)a n−1 + r 2 (n)a n−2 + · · · + r d (n)a n−d , where r i (n) are rational functions of n (see [7,Section 6.4]). Došlić and Veljan [5] presented a method on proving the log-convexity of the P-recursive sequences. Chen and Xia [3] gave a criterion for the 2-log-convexity of the P-recursive sequences of order 2.…”