# Category: Factorial and binomial topics

Factorial prime
A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (se
Beta distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] in terms of two positive parameters, denoted by alpha (α)
Table of Newtonian series
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence written in the form where is the binomial coefficient and is the falling factorial. Newtonian series often appear
Sun's curious identity
In combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002:
Factorial moment
In probability theory, the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments are useful for studying no
Gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distri
Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions whic
Mahler's theorem
In mathematics, Mahler's theorem, introduced by Kurt Mahler, expresses continuous p-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result: Let be the
Lobb number
In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced p
Doubly triangular number
In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if denotes the th triangula
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time o
Nørlund–Rice integral
In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory o
Generalized Pochhammer symbol
In mathematics, the generalized Pochhammer symbol of parameter and partition generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as It is used in multivariat
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively
De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna
De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna is a 38-page mathematical treatise written in the early 17th century by Thomas Harriot, lost for many years, and fin
Generalized hypergeometric function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a gener
Stirling transform
In combinatorial mathematics, the Stirling transform of a sequence { an : n = 1, 2, 3, ... } of numbers is the sequence { bn : n = 1, 2, 3, ... } given by where is the Stirling number of the second ki
Binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and
Vantieghems theorem
In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n(n≥3) is prime if and only if Similarly, n is prime, if and only if the following congruence for polyno
Wolstenholme's theorem
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one m
Fibonomial coefficient
In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as where n and k are non-negative integers, 0 ≤ k ≤ n, Fj is the j-th Fibonacci number and n!F is the nth Fib
Binomial regression
In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is the number of successes in a series of independe
Dyson conjecture
In mathematics, the Dyson conjecture (Freeman Dyson ) is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 by Wilson and Gunson. Andrews generalized it
Lozanić's triangle
Lozanić's triangle (sometimes called Losanitsch's triangle) is a triangular array of binomial coefficients in a manner very similar to that of Pascal's triangle. It is named after the Serbian chemist
Proof of Bertrand's postulate
In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The
Vandermonde's identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: for any nonnegative integers r, m, n. The identity is named after Alexandre
Legendre's formula
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes k
Difference polynomials
In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selb
Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also descri
Sperner's theorem
Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set
Hockey-stick identity
In combinatorial mathematics, the identity or equivalently, the mirror-image by the substitution : is known as the hockey-stick, Christmas stocking identity, boomerang identity, or Chu's Theorem. The
Hypergeometric distribution
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a
Dixon's identity
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of thr
Lah number
In mathematics, the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the th derivatives of .
Negative hypergeometric distribution
In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without replacement in which each sample can be classi
Sheffer sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence (pn(x) : n = 0, 1, 2, 3, ...) of polynomials in which the index of each polynomial equals its degree, satisfyi
Factorial number system
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not f
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word
Hyperfactorial
In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to .
Hypergeometric function
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or
Binomial series
In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the Taylor
Egorychev method
The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Cat
Bhargava factorial
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava a
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for eac
Falling and rising factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called th
Binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)
Extended negative binomial distribution
In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution. It is a truncated version of the negative
Hypergeometric identity
In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in soluti
Multinomial distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided dice rolled n times. F
Wilson prime
In number theory, a Wilson prime is a prime number such that divides , where "" denotes the factorial function; compare this with Wilson's theorem, which states that every prime divides . Both are nam
Negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distribu
Double factorial
In mathematics, the double factorial or semifactorial of a number n, denoted by n‼, is the product of all the integers from 1 up to n that have the same parity (odd or even) as n. That is, For even n,
Binomial transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is th
Fuss–Catalan number
In combinatorial mathematics and statistics, the Fuss–Catalan numbers are numbers of the form They are named after N. I. Fuss and Eugène Charles Catalan. In some publications this equation is sometime
Quasibinomial
No description available.
Factorial moment generating function
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as for all complex numbers t for whi
Brocard's problem
Brocard's problem is a problem in mathematics that asks to find integer values of and for which where is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885, and indepe
0!
No description available.
Central binomial coefficient
In mathematics the nth central binomial coefficient is the particular binomial coefficient They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triang
Catalan number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the Frenc
Hermite interpolation
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a
Bernoulli's triangle
Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0 and n, the component in row n and column k is giv
In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the t
MacMahon's master theorem
In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph Combinatory analysis (1916)
Eulerian number
In combinatorics, the Eulerian number A(n, m) is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). The
Sierpiński triangle
The Sierpiński triangle (sometimes spelled Sierpinski), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subd
Finite difference
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by fini
Macaulay representation of an integer
Given positive integers and , the -th Macaulay representation of is an expression for as a sum of binomial coefficients: Here, is a uniquely determined, strictly increasing sequence of nonnegative int
Generalized integer gamma distribution
In probability and statistics, the generalized integer gamma distribution (GIG) is the distribution of the sum of independent gamma distributed random variables, all with integer shape parameters and
Factorial
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller
N! conjecture
In mathematics, the n! conjecture is the conjecture that the dimension of a certain module of is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's pos
Gaussian binomial coefficient
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomi
Rook polynomial
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in
Alternating factorial
In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers. This is the same as their sum, with the odd-indexed factorials mul
Singmaster's conjecture
Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on th
Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers in which the index of each polynomial equals its degree, is said to be of binomial type if it sa
Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is
Kempner function
In number theory, the Kempner function is defined for a given positive integer to be the smallest number such that divides the factorial . For example, the number does not divide , , or , but does div
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno , although he was not the first to state or prove the f
Pascal's simplex
In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.
Beta negative binomial distribution
In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable equal to the number of failures needed to get successes in a sequence of inde
Stirling number
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n
Multiplicative partitions of factorials
Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied by Paul Erdős and others. The factorial of a
Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according t
Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of sparse polynomial after the monomials.
Genocchi number
In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation The first few Genocchi numbers are 0, −1, −1, 0, 1, 0, −3, 0, 17 (sequence in
Jordan–Pólya number
In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, is a Jordan–Pól
Pochhammer k-symbol
In the mathematical theory of special functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan are generalizations of the Pochhammer symbol and gamma fu
Binomial approximation
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that It is valid when and where and may be real or complex numbers. The benefit o
Triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n
List of factorial and binomial topics
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). * Abel's binomial theorem * Alternating factorial * Antichain * Beta function * Bhargava factor
Combinatorial number system
In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an int
Exponential factorial
The exponential factorial is a positive integer n raised to the power of n − 1, which in turn is raised to the power of n − 2, and so on and so forth in a right-grouping manner. That is, The exponenti
Trinomial expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by where n is a nonnegative integer and the sum is taken over all combi
Rothe–Hagen identity
In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers except where its denominators vanish: It is a generalization of Vandermonde's identity, and is named a
Negative multinomial distribution
In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes. As with the univariate nega
Pillai prime
In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a mul
Newton polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is som
Poisson binomial distribution
In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distribut
Abel's binomial theorem
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following:
Newton–Pepys problem
The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice. In 1693 Samuel Pepys and Isaac Newton corresponded over a problem posed to
Narayana number
In combinatorics, the Narayana numbers form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T.
Bernoulli umbra
In Umbral calculus, Bernoulli umbra is an , a formal symbol, defined by the relation , where is the index-lowering operator, also known as evaluation operator and are Bernoulli numbers, called moments
Superfactorial
In mathematics, and more specifically number theory, the superfactorial of a positive integer is the product of the first factorials. They are a special case of the Jordan–Pólya numbers, which are pro
Star of David theorem
The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972.
Trinomial triangle
The trinomial triangle is a variation of Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the three (rather than the two in Pascal's triangle)
Binomial distribution
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments,
Multinomial theorem
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multino
Fox–Wright function
In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) bas
Gould's sequence
Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only of powers of two, and begins: 1, 2, 2, 4, 2,
Pascal's pyramid
In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is