Related papers: On the log-concavity of $n$-th root of a sequence

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…

Recently, Z. W. Sun put forward a series of conjectures on monotonicity of combinatorial sequences in the form of $\{z_n/z_{n-1}\}_{n=N}^\infty$ and $\{\sqrt[n+1]{z_{n+1}}/\sqrt[n]{z_n}\}_{n=N}^\infty$ for some positive integer $N$, where…

In his study of Ramanujan-Sato type series for $1/\pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials…

Let $\{P_n\}_{n\geq 0}$ denote the Catalan-Larcombe-French sequence, which naturally came up from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence…

Recently, Z. W. Sun introduced a new kind of numbers $S_n$ and also posed a conjecture on ratio monotonicity of combinatorial sequences related to $S_n$. In this paper, by investigating some arithmetic properties of $S_n$, we give an…

Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)>\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\sqrt[n]{p(n)/n}$ and…

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…

Towards confirming Sun's conjecture on the strict log-concavity of combinatorial sequence involving the n$th$ Bernoulli number, Chen, Guo and Wang proposed a conjecture about the log-concavity of the function…

Recently, Z. W. Sun introduced a sequence $(S_n)_{n\geq 0}$, where $S_n=\frac{\binom{6n}{3n} \binom{3n}{n}}{2(2n+1)\binom{2n}{n}}$, and found one congruence and two convergent series on $S_n$ by {\tt{Mathematica}}. Furthermore, he proposed…

We consider the higher order Tur\'an inequality and higher order log-concavity for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}}…

We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form $(\root n\of{a_n})_{n\ge 1}$ or the form $(\root{n+1}\of{a_{n+1}}/\root n\of{a_n})_{n\ge1}$, where $(a_n)_{n\ge 1}$ is a…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if $\mathcal{L}^j (a_n) \geq 0$ for all $j = 0, 1, \ldots, m$. Here, $\mathcal{L}$ is the operator defined by $\mathcal{L} (a_n) = a_n^2 - a_{n - 1} a_{n + 1}$. By a criterion of…

We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.

Sun, in 2022, introduced a conjectured evaluation for a series of convergence rate $\frac{1}{2}$ involving harmonic numbers. We prove both this conjecture and a stronger version of this conjecture, using a summation technique based on a…

Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) "infinitely…

This paper presents the log-concavity of the $m$-gonal figurate number sequences. The author gives and proves the recurrence formula for $m$-gonal figurate number sequences and its corresponding quotient sequences which are found to be…

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed…

Let $A_n=(a_0,a_1,\dots,a_{n-1})$ be drawn uniformly at random from $\{-1,+1\}^n$ and define \[ M(A_n)=\max_{0<u<n}\,\Bigg|\sum_{j=0}^{n-u-1}a_ja_{j+u}\Bigg|\quad\text{for $n>1$}. \] It is proved that $M(A_n)/\sqrt{n\log n}$ converges in…

Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset…