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Swinging Factorial , User:Peter Luschny/Multifactorials

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1.1A conjecture of Erdös, Graham et al. 1.2Concerned with sequences 1.3Computing the sequences 1.4Discoveries 1.5References On the prime factors of the swinging factorial. KEYWORDS: swinging factorial, central binomial coefficient, conjecture of Erdös, Graham et al.

A sequence transformation and the Bernoulli numbers Peter Luschny, April 2010 KEYWORDS: Bernoulli, Euler, Tangent, Harmonic, Binomial, Swiss-Knife, Worpitzky, Akiyama Fatorial oscilante factorial swinging 2. Простые множители swinging factorial Обозначим как степень простого числа в примарном разложении . Тогда будет справедлива следующая формула: Доказательство

Double Factorials - Robert Dickau

This document provides an overview of a new divide-and-conquer algorithm called the dsc-factorial for efficiently computing the factorial function n!. It describes: 1) A decomposition of the factorial function n! into a „swinging factorial“ no and another oscillating function, allowing n! to be computed recursively using no. 2) A theorem showing the prime factors of no can be end return ∏(factors) end doc“““ Computes the swinging factorial (a.k.a. swing numbers n≀). Cf. A056040. „““ function swing(n::Int) sh = count_ones(div(n,2)) swing_oddpart(n) << sh end const FactorialOddPart = [1, 1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 638512875, 638512875, 10854718875, 97692469875,

User:Peter Luschny/Multifactorials

I found FastFactorialFunctions describing a number of algorithms for computing the factorial. Unfortunately, the explanations are terse and I don’t feel like sifting through line after line of source code to understand the basic principles behind the algorithms. Can anybody point me to more detailed descriptions of these (or other fast) algorithms for computing large exact

La mejor manera de crear un portal del empleado para tu empresa sin programar con Factorial sistema de recursos humanos 100% segura y gratis. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give Hints

Nombres, curiosités, théorie et usages: factorielles oscillantes, swinging factorials A geometrical approach of the lonely runner conjecture Didier Guillet _27/06/2018 La conjecture du coureur solitaire peut être exprimée sous une forme géométrique. Les n coureurs sont remplacés par un réseau d’ hyper-cubes dans un espace de dimension n-1.

424: var swing = combi.Swing (1000); 425: System.Console.WriteLine („Swing: “ + swing); 426: 427: var fact = combi.Factorial (1000); 428: System.Console.WriteLine („Factorial: “ + fact); 429: 430: var dblfact = combi.DoubleFactorial (1000); 431: System.Console.WriteLine („DoubleFactorial: “ + dblfact); 432: 433: var pascal = combi.Binomial Permutation Trees Peter Luschny, 2010-07-20 Contents * 1 Permutation Trees o 1.1 Definitions o 1.2 Example classifications * 2 Permutation trees with power 4 * 3 Counting permutation trees o 3.1 A008292 Permutation trees of power n and width k. o 3.2 A179454 Permutation trees of power n and height k. o 3.3 A179455 Permutation trees of power n and FFF CompLang Shootout

Fast Factorial Functions. [fff] Ein Kampf gegen die Dummheit, die Fakultät als n! = n* (n-1)! zu berechnen. Diese Seite ist im berühmten „Dictionary of Algorithms and Data Structures“ des amerikanischen National Institute of Standards and Technology (NIST) aufgeführt und ist die am meisten besuchte Seite meiner Homepage.

n! gives the factorial of n.Factorial represents the factorial function. In particular, Factorial [n] returns the factorial of a given number , which, for positive integers, is defined as . For n 1,2,, the first few values are therefore 1,2,6,24,120,720,. The special case is defined as 1, consistent with the combinatorial interpretation of there being exactly one way to arrange zero

Irregular Bernoulli and Euler Primes

The swinging factorial. What is the swinging factorial of 5? – YouTube About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety Factorial function, Swinging factorial, Gamma function, Era-tosthenes’ factorial, Binomial coefficient, Least common multiple, Cata-lan numbers, Louisa numbers, Combinatorial pyramids, Swinging orbi-tals, Bertrand’s postulate, Alhacen-Lagrange theorem, Riemann hypothe-ses, Hypergeometric series, Reed-Dawson identity, Orbital lattice, Omega

The relations between Factorial(k,n), Variation(k,n) and Arrangement(k,n). The connection between a generalized factorial function and a generalization of the sequence of variations and the sequence of arrangements is the most important relation considered here. Let us review it in a systematic framework. First we define the function Качающийся факториал (англ. swinging factorial) — функция, определённая на множестве неотрицательных целых чисел Z + {displaystyle mathbb {Z} ^ {+}} . 1. Factorial decomposition We introduce a function called swinging factorial , as follows:

Hier sollte eine Beschreibung angezeigt werden, diese Seite lässt dies jedoch nicht zu.

The OEIS is supported by the many generous donors to the OEIS Foundation. Perfect and Optimal Rulers Generating and counting. What is a perfect ruler? A short introduction. In English, in German, in French and in Russian. The basic definitions: Rulers which are complete can have nice properties like being length-maximal or marker-minimal. A more formal description. Counting rulers: All checked values known. Here is a table, some samples and the perfect ruler Write a function which generates the swinging factorial for a given positive integer, n. All of the tests inputs will be valid, so there’s no need to validate them.

We compute a natural log limit where our input consist of the swinging factorial.Thanks for watching. Like, comment, and subscribe for more Fast Factorial Functions. [fff] Ein Kampf gegen die Dummheit, die Fakultät als n! = n* (n-1)! zu berechnen. Diese Seite ist im berühmten „Dictionary of Algorithms and Data Structures“ des amerikanischen National Institute of Standards and Technology (NIST) aufgeführt und ist die am meisten besuchte Seite meiner Homepage.

On swinging factorials and the

Introduction You’ve probably heard of factorials and there’s a chance you’ve written, or attempted, to create some code which can calculate them. There is a lesser known variant of factorials kn

The Binomial (n over k) The binomial function computes the binomial coefficients. If the arguments are both positive integers with 0 <= k <= n, then binomial (n, k) = n! / (k! * (n-k)!) which is the number of ways of choosing k objects from n distinct objects. Otherwise an exception is thrown. For each prime p <= n, we compute the power of p e dividing the binomial coefficient KEYWORDS: Irregular Primes, Bernoulli numbers, Euler numbers, computing modulo p, Akiyama-Tanigawa algorithm, Ernst Eduard Kummer, Thomas Mautsch, OEIS A000928, OEIS A120337, OEIS A128197. "The pioneering mathematician Kummer, over the period 1847-1850, used his profound theory of cyclotomic fields to establish a certain class of primes called Our goal is to define a triangle of binomial coefficients where the swinging factorial is always the middle coefficient, and not only in the case when $n$ is even.

Теперь, зная степени всех простых делителей n≀, у нас есть способ вычисления swinging factorial. С какой сложностью нам достается это знание?

The swinging factorial can be generalized to a complex function very similar like the factorial function can be generalized to the complex Gamma function. We will not further pursue this subject here as we have promised to look at the Catalan numbers.

%N Swinging factorial, a (n) = 2^ (n- (n mod 2))*Product_ {k=1..n} k^ ( (-1)^ (k+1)). %C a (n) is the number of ’swinging orbitals‘ which are enumerated by the trinomial n over [floor (n/2), n mod 2, floor (n/2)]. Fast Double Factorial Functions ‼ Definitions and formulas. The name ‚doublegamma‘ is ad hoc. If you ever wonder where the name ’swing‘ in the text below comes from this plot might give you a clue. The factorial world behind the standard factorial is no longer monotonic! Looking only at the values of integers will deceive you‼ log (abs (gamma)) (red) versus log (abs (doublegamma)) The factorial numbers can be calculated by means of the recurrence n! = (floor (n/2)!)^2 * sf (n) where sf (n) are the swinging factorials A056040. This leads to

Perfect and Optimal Rulers Generating and counting. What is a perfect ruler? A short introduction. In English, in German, in French and in Russian. The basic definitions: Rulers which are complete can have nice properties like being length-maximal or marker-minimal. A more formal description. Counting rulers: All checked values known. Here is a table, some samples and the perfect ruler

A swinging factorial limit

The Binomial (n over k) The binomial function computes the binomial coefficients. If the arguments are both positive integers with 0 <= k <= n, then binomial (n, k) = n! / (k! * (n-k)!) which is the number of ways of choosing k objects from n distinct objects. Otherwise an exception is thrown. For each prime p <= n, we compute the power of p e dividing the binomial coefficient binomial