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# Binomial coefficient

In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. [1]

122 relations: Almost all, American Mathematical Monthly, American Mathematical Society, Andreas von Ettingshausen, Andrew Granville, APL (programming language), Axiom of choice, Beta function, Bhāskara II, Binary entropy function, Binary number, Binomial (polynomial), Binomial distribution, Binomial theorem, Binomial transform, Bit, C (programming language), Cardinal number, Cardinality, Catalan number, Central binomial coefficient, Characteristic (algebra), Coefficient, Combination, Combinatorial proof, Combinatorics, Commutative ring, Computer terminal, David Singmaster, Delannoy number, Derivative, Differential equation, Dixon's identity, Double counting (proof technique), Ernst Kummer, Euler's formula, Euler–Mascheroni constant, Eulerian number, Exponentiation, Factorial, Falling and rising factorials, Fibonacci number, Field (mathematics), Finite difference, Formal power series, Fractional part, Gamma function, Gaussian binomial coefficient, Generating function, German tank problem, ... Expand index (72 more) »

## Almost all

In mathematics, the term "almost all" means "all but a negligible amount".

## American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Andreas von Ettingshausen

Andreas Freiherr von Ettingshausen (25 November 1796 – 25 May 1878) was a German mathematician and physicist.

## Andrew Granville

Andrew James Granville (born 7 September 1962) is a British mathematician, working in the field of number theory.

## APL (programming language)

APL (named after the book A Programming Language) is a programming language developed in the 1960s by Kenneth E. Iverson.

## Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

## Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by for.

## Bhāskara II

Bhāskara (also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhaskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer.

## Binary entropy function

In information theory, the binary entropy function, denoted \operatorname H(p) or \operatorname H_\text(p), is defined as the entropy of a Bernoulli process with probability p of one of two values.

## Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).

## Binomial (polynomial)

In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial.

## 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, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q.

## Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

## Binomial transform

In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences.

## Bit

The bit (a portmanteau of binary digit) is a basic unit of information used in computing and digital communications.

## C (programming language)

C (as in the letter ''c'') is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations.

## Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Catalan number

In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects.

## 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 triangle.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

## Combination

In mathematics, a combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter.

## Combinatorial proof

In mathematics, the term combinatorial proof is often used to mean either of two types of mathematical proof.

## Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

## Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

## Computer terminal

A computer terminal is an electronic or electromechanical hardware device that is used for entering data into, and displaying or printing data from, a computer or a computing system.

## David Singmaster

David Breyer Singmaster (born 1939, USA) is a retired professor of mathematics at London South Bank University, England, UK.

## Delannoy number

In mathematics, a Delannoy number D describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## 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 three binomial coefficients, and some evaluating a hypergeometric sum.

## Double counting (proof technique)

In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set.

## Ernst Kummer

Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.

## Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

## Euler–Mascheroni constant

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

## 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").

## Exponentiation

Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n.

## Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Finite difference

A finite difference is a mathematical expression of the form.

## Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.

## Fractional part

The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part.

## Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## 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.

## Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.

## German tank problem

In the statistical theory of estimation, the German tank problem consists in estimating the maximum of a discrete uniform distribution from sampling without replacement.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

## Halayudha

Halayudha (Sanskrit: हलायुध) was a 10th-century Indian mathematician who wrote the, a commentary on Pingala's Chandaḥśāstra.

## Harmonic number

In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.

## Hockey-stick identity

In combinatorial mathematics, the identity is known as the hockey-stick or Christmas stocking identity.

## 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 limiting cases.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integer-valued polynomial

In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.

## Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

## J (programming language)

The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is a synthesis of APL (also by Iverson) and the FP and FL function-level languages created by John Backus.

## Java (programming language)

Java is a general-purpose computer-programming language that is concurrent, class-based, object-oriented, and specifically designed to have as few implementation dependencies as possible.

## Julia (programming language)

Julia is a high-level dynamic programming language designed to address the needs of high-performance numerical analysis and computational science, without the typical need of separate compilation to be fast, while also being effective for general-purpose programming, web use or as a specification language.

## Kummer's theorem

In mathematics, Kummer's theorem for binomial coefficients gives the ''p''-adic valuation of a binomial coefficient, i.e., the exponent of the highest power of a prime number p dividing this binomial coefficient.

## Līlāvatī

The Līlāvatī is Indian mathematician Bhāskara II's treatise on mathematics, written in 1150.

## List of factorial and binomial topics

This is a list of factorial and binomial topics in mathematics.

## Logarithmic differentiation

In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself.

## Lucas's theorem

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient \tbinom by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.

## Mathematical Association of America

The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mathematics Magazine

Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

## Maxima (software)

Maxima is a computer algebra system (CAS) based on a 1982 version of Macsyma.

## Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.

## Motzkin number

In mathematics, a Motzkin number for a given number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord).

## 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.

## Multiset

In mathematics, a multiset (aka 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.

## Narayana number

In combinatorics, the Narayana numbers N(n, k), n.

## Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## 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.

## Neurocomputing (journal)

Neurocomputing publishes articles in the field of neural networks and machine learning.

## 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.

## PARI/GP

PARI/GP is a computer algebra system with the main aim of facilitating number theory computations.

## Partial fraction decomposition

In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

## Pascal's rule

In mathematics, Pascal's rule is a combinatorial identity about binomial coefficients.

## Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

## Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

## Pingala

Pingala (Devanagari: पिङ्गल) (c. 3rd/2nd century BC) was an ancient Indian mathematician who authored the (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Polynomial expansion

In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## Programming language

A programming language is a formal language that specifies a set of instructions that can be used to produce various kinds of output.

## Python (programming language)

Python is an interpreted high-level programming language for general-purpose programming.

## Q-analog

In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as.

## R (programming language)

R is a programming language and free software environment for statistical computing and graphics that is supported by the R Foundation for Statistical Computing.

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Recursion

Recursion occurs when a thing is defined in terms of itself or of its type.

## Sanskrit

Sanskrit is the primary liturgical language of Hinduism; a philosophical language of Hinduism, Sikhism, Buddhism and Jainism; and a former literary language and lingua franca for the educated of ancient and medieval India.

## Scheme (programming language)

Scheme is a programming language that supports multiple paradigms, including functional programming and imperative programming, and is one of the two main dialects of Lisp.

## SciPy

SciPy (pronounced /ˈsaɪpaɪ'/ "Sigh Pie") is a free and open-source Python library used for scientific computing and technical computing.

## Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

## Series multisection

In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series.

## Singmaster's conjecture

Singmaster's conjecture is a conjecture in combinatorial number theory in mathematics, named after the British mathematician David Singmaster who proposed it in 1971.

## Society for Industrial and Applied Mathematics

The Society for Industrial and Applied Mathematics (SIAM) is an academic association dedicated to the use of mathematics in industry.

## Star of David theorem

The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients.

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

## Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.

## Stirling's approximation

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials.

## String (computer science)

In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable.

## Subroutine

In computer programming, a subroutine is a sequence of program instructions that performs a specific task, packaged as a unit.

## 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: (x+m+1)\sum_^m(-1)^i\dbinom\dbinom -\sum_^\dbinom(-4)^i.

## Table of Newtonian series

In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form where is the binomial coefficient and (s)_n is the rising factorial.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Trinomial expansion

In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials.

## Typewriter

A typewriter is a mechanical or electromechanical machine for writing characters similar to those produced by printer's movable type.

## 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-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie (Chu Shi-Chieh).

## Wolfram Mathematica

Wolfram Mathematica (usually termed Mathematica) is a modern technical computing system spanning most areas of technical computing — including neural networks, machine learning, image processing, geometry, data science, visualizations, and others.

## Zeckendorf's theorem

Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.

## References

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