scholarly journals On some results concerning the polygonal polynomials

2019 ◽  
Vol 35 (1) ◽  
pp. 01-12
Author(s):  
DORIN ANDRICA ◽  
◽  
OVIDIU BAGDASAR ◽  
Keyword(s):  
Recurrence Relations ◽  
Integer Sequences ◽  
Open Questions ◽  
New Meanings ◽  
Online Encyclopedia ◽  
Integral Formulae

In this paper we define the nth polygonal polynomial and we investigate recurrence relations and exact integral formulae for the coefficients of Pn and for those of the Mahonianpolynomials. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

scholarly journals Truncated-Exponential-Based Appell-Type Changhee Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1588
Author(s):  
Tabinda Nahid ◽  
Parvez Alam ◽  
Junesang Choi
Keyword(s):  
Recurrence Relations ◽  
Explicit Representation ◽  
Integral Inequalities ◽  
Exponential Polynomials ◽  
Integral Formulas ◽  
Open Questions ◽  
Changhee Polynomials ◽  
Natural Way ◽  
Addition Formulas

The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials Chn*(x) (10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials eCn*(x) in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials eCn*(x). Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested.

scholarly journals Polygonal Numbers

2013 ◽  
Vol 21 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Related Result ◽  
Integer Sequences ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Polygonal Numbers ◽  
Online Encyclopedia ◽  
Four Square ◽  
Mersenne Primes ◽  
Ordinal Index

Summary In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = Δ+Δ+Δ”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers. We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217). An interesting related result [16] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [32].

scholarly journals A New Record of Graph Enumeration Enabled by Parallel Processing

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1214 ◽  
Author(s):  
Zhipeng Xu ◽  
Xiaolong Huang ◽  
Fabian Jimenez ◽  
Yuefan Deng
Keyword(s):  
Parallel Processing ◽  
Shortest Path ◽  
Interconnection Networks ◽  
New Record ◽  
Parallel Computers ◽  
Regular Graphs ◽  
Integer Sequences ◽  
Path Lengths ◽  
Online Encyclopedia ◽  
Processor Cores

Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in the Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular graphs of order 23 as 429,668,180,677,439, while discovering several regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The enumeration of 4-regular graphs and the discovery of minimal-ASPL graphs are extremely time consuming. We accomplish them by adapting GENREG, a classical regular graph generator, to three supercomputers with thousands of processor cores.

scholarly journals On Partially Ordered Patterns of Length 4 and 5 in Permutations

10.37236/8605 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Alice L. L. Gao ◽  
Sergey Kitaev
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Relative Order ◽  
Permutation Patterns ◽  
Integer Sequences ◽  
Arbitrary Length ◽  
Partially Ordered ◽  
Long Line ◽  
Online Encyclopedia ◽  
Wilf Equivalence

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length $k$ is defined by a partially ordered set on $k$ elements, and classical patterns correspond to $k$-element chains. The notion of a POP provides  a convenient language to deal with larger sets of permutation patterns. This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain  13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

scholarly journals New families of special numbers for computing negative order Euler numbers and related numbers and polynomials

2018 ◽  
Vol 12 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Bernstein Polynomials ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Euler Numbers ◽  
Open Questions ◽  
Negative Order ◽  
Central Factorial Numbers ◽  
Probability And Statistics

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

scholarly journals A note on the Neyman–Rayner triangle

2021 ◽  
Vol 27 (4) ◽  
pp. 164-166
Author(s):  
A. G. Shannon ◽  
Keyword(s):  
Legendre Polynomials ◽  
Cluster Algebra ◽  
Integer Sequences ◽  
Statistical Research ◽  
Orthonormal Polynomials ◽  
Online Encyclopedia

This note raises questions for other number theorists to tackle. It considers a triangle arising from some statistical research of John Rayner and his use of some orthonormal polynomials related to the Legendre polynomials. These are expressed in a way that challenges the generalizing them. In particular, the coefficients are expressed in a triangle and related to known sequences in the Online Encyclopedia of Integer Sequences. The note actually raises more questions than it answers when it links with the cluster algebra of Fomin and Zelevinsky.

Integer sequences that behave as Fibonacci-Lucas pairs

2013 ◽  
Vol 97 (538) ◽  
pp. 1-7
Author(s):  
A. S. Di Domenico
Keyword(s):  
Recurrence Relations ◽  
Second Order ◽  
Integer Sequences ◽  
Hyperbolic Functions ◽  
Lucas Sequences ◽  
Constant Coefficients ◽  
Image Position

There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1]. This article is based on other integer sequences. It sets out to determine other pairs of such sequences that have the same relation as the Fibonacci and Lucas have to each other. So we shall be concerned with second order recurrence relations with constant coefficients:and pairs of sequences (un) and (vn) that each satisfy it. We seek a condition that ensures the pair of sequences behave as the Fibonacci-Lucas pair behave.

scholarly journals The Number of Ways to Assemble a Graph

10.37236/2644 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Andrew Vince ◽  
Miklós Bóna
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Explicit Formulas ◽  
Future Directions ◽  
Macromolecular Assembly ◽  
Open Questions ◽  
Asymptotic Formulae ◽  
Special Cases ◽  
Assembly Problem ◽  
Precise Asymptotic

Motivated by the question of how macromolecules assemble,the notion of an assembly tree of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,\phi)$-graphs.  In some natural special cases, we use a powerful recent result of Zeilberger and Apagodu to provide recurrence relations for the diagonal of the relevant multivariate generating functions, and we use a result of Wimp and Zeilberger to find very precise asymptotic formulae for the coefficients of these diagonals.  Future directions for reseach, as well as open questions, are suggested.

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