Log-balanced combinatorial sequences

In recent years, Sun has proposed numerous conjectures regarding the log-concavity of root sequences $\{\sqrt[n]{a_n}}_{n\geqslant 1}$. We establish criteria for the asymptotic log-concavity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$ and the asymptotic ratio log-convexity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$ for $P$-recursive sequences $\{\sqrt[n]{a_n}}_{n\geqslant{0}}$. Additionally, by the aid of symbolic computation, we present a systematic approach to determine the explicit int...

We numerically estimate the critical exponents of certain enumeration sequences that naturally generalize the famous Catalan and super-Catalan sequences, and raise deep and original questions about their exact values, and whether they are rational numbers. In this version we announce that our questions were brilliantly answered by Michael Wallner, and that the pledged donation to the OEIS, in his honor, was made.

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.

We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of logarithmic intertwining operators and logarithmic tensor category theory for modules for a vertex operator algebra. This extension has a variety of interesting arithmetic properties. We develop one such result here, the aforementioned recursive iden...

We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial objects, e.g., undirected graphs, permutations, matchings etc. In the second step, we use algorithms for lifted first-order model counting to generate integer sequences that count the objects encoded by the first-order logic formulas generated ...

Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers a(n), but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) st...

The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\sqrt[x]{f(x)}$ is given, which is a continuous analog for one result of Wang and Zhu. (2) The log-behavior of the functions $\theta(x)=\sqrt[x]{2 \zeta(x)\Gamma(x+1)}$ and $F(x)=\sqrt[x]{\frac{\Gamma(ax+b+1)}{\Gamma(c x+d+1)\Gamma(e x+f+1)}}$ is considered, where $\zeta(x)$ and $\Gamma(x)$ are the Riemann ...

A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.

This paper has been withdrawn by the author; a revised version is part of the author's phd-thesis "Quasi-logarithmic structures" (Zurich, 2007).

The paper investigates the properties of a nonlinear recursive sequence which includes several ones studied formerly in the literature.