Integer sequences

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 Schr\"oder numbers, sequences of values of some classic orthogonal polynomials, and many others. The calculus method extends also to two- (or more-) indexed sequences.

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the general Eulerian polynomials, as defined by Xiong, Tsao and Hall, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of this polynomial sequence in terms of continued fractions, and we also calculate the Hankel transforms of the polynomial sequence. We indicate that the polynomial sequence can be characterized by the further notion of generalized Eulerian distribution first introduced by Morisita. We finish with examples of related Pascal-like triangles.

The authors study the global and local behaviour of the truncated kernel function γ 2 (n)=γ(n) p(n), where γ(n)=∏ p|n p is the kernel function and P(u) the largest prime factor of n. They prove asymptotic formulas of ∑ n≤x γ 2 (n) and ∑ n≤x 1 γ 2 (n). The authors further show, that given any positive real n<1 lim x→∞ 1 x#{n≤x:γ 2 (n)<x n }=lim x→∞ 1 x#n ≤ x : n p(n) < x n =1-ρ1 1-μ, where p is the Dickman function. – Finally, they prove that n P(n) is very often much larger than γ 2 (n).Reviewer: Jürgen Spilker (Freiburg i. Br.)

What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various times throughout history, one was often not the first positive integer (some started with two, and a few with three). In this article, we survey the history of the primality of one, from the ancient Greeks to modern times. We will discuss some of the reasons definitions changed, and provide several examples. We will also discuss the last significant mathematicians to list the number one as prime.

In this paper, we show the internal relations among the elements of the circular sequence (1,12,21,123,231,312,1234,2341,…). We illustrate one method to minimize the number of the “candidate prime numbers” up to a given term of the sequence. So, having chosen a particular prime divisor, it is possible to analyze only a fixed number of the smallest terms belonging to a given range, thus providing the distribution of that prime factor in a larger set of elements. Finally, we combine these results with another one, also expanding the study to a few new integer sequences related to the circular one.

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers ! Q,n counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function ! Q(x) = ! ! Q,nx n and use it to show that ! 1/n Q,n tends to Q 1/(p! 1) . We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture contrasts sharply with known results for function fields, and our arguments explain this contrast.

It is well known that the numbers $(2m)! (2n)!/m! n! (m+n)!$ are integers, but in general there is no known combinatorial interpretation for them. When $m=0$ these numbers are the middle binomial coefficients $\binom{2n}{n}$, and when $m=1$ they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers when $m=2$ or 3.

We generalize well-known Catalan-type integrals for Euler's constant to values of the generalized-Euler-constant function and its derivatives. Using generating functions appeared in these integral representations we give new Vacca and Ramanujan-type series for values of the generalized-Euler-constant function and Addison-type series for values of the generalized-Euler-constant function and its derivative. As a consequence, we get base $B$ rational series for $\log\frac{4}{\pi},$ $\frac{G}{\pi}$ (where $G$ is Catalan's constant), $\frac{\zeta'(2)}{\pi^2}$ and also for logarithms of Somos's and Glaisher-Kinkelin's constants.

We investigate a coin-weighing puzzle that appeared in the all-Russian math Olympiad in 2000. We liked the puzzle because the methods of analysis differ from classical coin-weighing puzzles. We generalize the puzzle by varying the number of participating coins, and deduce a complete solution, perhaps surprisingly, the objective can be achieved in no more than two weighings regardless of the number of coins involved.

This paper is devoted to the study of eigen-sequences for some important operators acting on sequences. Using functional equations involving generating functions, we completely solve the problem of characterizing the fixed sequences for the Generalized Binomial operator. We give some applications to integer sequences. In particular we show how we can generate fixed sequences for Generalized Binomial and their relation with the Worpitzky transform. We illustrate this fact with some interesting examples and identities, related to Fibonacci, Catalan, Motzkin and Euler numbers. Finally we find the eigen-sequences for the mutual compositions of the operators Interpolated Invert, Generalized Binomial and Revert.

In this paper, we show the internal relations among the elements of the circular sequence (1,12,21,123,231,312,1234,2341,…). We illustrate one method to minimize the number of the “candidate prime numbers ” up to a given term of the sequence. So, having chosen a particular prime divisor, it is possible to analyze only a fixed number of the smallest terms belonging to a given range, thus providing the distribution of that prime factor in a larger set of elements. Finally, we combine these results with another one, also expanding the study to a few new integer sequences related to the circular one.

We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized Euler constant function and its derivatives. Using generating functions appearing in these integral representations, we give new Vacca and Ramanujan-type series for values of the generalized Euler constant function and Addison-type series for values of the generalized Euler constant function and its derivative. As a consequence, we get base-B rational series for log 4 G π π (where G is Catalan’s constant), ζ ′ (2) π2 and also for logarithms of the Somos and Glaisher-Kinkelin constants. 1

In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation.

A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1,1) and (1, −1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0,0) and finish at (2n,0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice points p1 and p2, for the number of Dyck paths starting at p1, ending at p2, and touching the x-axis exactly t times. Moreover, we provide explicit expressions for the corresponding generating function and bivariate generating function.

Using the language of Riordan arrays, we define a notion of generalized Bernstein polynomials which are defined as elements of certain Riordan arrays. We characterize the general elements of these arrays, and examine the Hankel transform of the row sums and the first columns of these arrays. We propose conditions under which these Hankel transforms possess the Somos-$4$ property. We use the generalized Bernstein polynomials to define generalized B\'ezier curves which can provide a visualization of the effect of the defining Riordan array.

The aim of this paper is to present a method of generating inequalities and, under certain conditions, some identities with sums that involve floor, ceiling and round functions. We apply this method to sequences of nonnegative integers that could be turned into periodical sequences.

We present techniques for obtaining a generating function for the central coefficients of a triangle $T(n,k)$, which is given by the expression $[xH(x)]^k=\sum_{n\geqslant k} T(n,k)x^n$, $H(0)\neq 0$. We also prove certain theorems for solving direct and inverse problems.

The present paper studies the diophantine equation GnHn + c = x2n and related questions, where the integer binary recurrence sequences {G}, {H} and {x} satisfy the same recurrence relation, and c is a given integer. We prove necessary and sufficient conditions for the solubility of GnHn + c = x2n. Finally, a few relevant examples are provided.

In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of character- istic polynomials are changed and we prove that a subset of these operators form a group, with respect to a well–defined composition law. Furthermore, we study a vast class of linear recurrent sequences fixed by these operators and many other interesting properties. Finally, we apply all the results to integer sequences, finding many rela- tions and formulas involving Catalan numbers, Fibonacci numbers, Lucas numbers and triangular numbers.

We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-tiling of length n