List Of Factorial And Binomial Topics

{{Short description, none This is a list of factorial and binomial topics in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. See also

binomial (disambiguation) Binomial may refer to: In mathematics *Binomial (polynomial), a polynomial with two terms *Binomial coefficient, numbers appearing in the expansions of powers of binomials * Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition ...

. * Abel's binomial theorem *

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 multiplied by −1 if ''n'' is even, and t ...

Antichain In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its wid ...

Beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...

* Bhargava factorial *

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 and is written \tbinom. It is the coefficient of the t ...

Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

Binomial distribution In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...

Binomial proportion confidence interval In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion conf ...

* Binomial-QMF ( Daubechies wavelet filters) *

Binomial series In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where \alpha is any complex number, and the power series on the right-hand side is expressed in terms of the ...

Binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...

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 the result of applying the binomial transform to ...

Binomial type In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities : ...

* Carlson's theorem *

Catalan number The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were p ...

** Fuss–Catalan number *

Central binomial coefficient In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first ...

Combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are ...

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 integer, is a correspondence between natura ...

De Polignac'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 known as de Polignac's formula, a ...

Difference operator In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

* Difference polynomials *

Digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...

* Egorychev method * Erdős–Ko–Rado theorem *

Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...

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 , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...

Factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

* Factorial moment *

Factorial number system In combinatorics, the factorial number system (also known as factoradic), is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, but as place value of ...

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 : : 2 (0! + 1 or 1! + 1) ...

* Factoriangular number *

Gamma distribution In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the g ...

Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

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 binomial coefficient, written as \binom nk ...

* Gould's sequence * Hyperfactorial *

Hypergeometric distribution In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the ...

* Hypergeometric function identities *

Hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...

* Incomplete beta function *

Incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...

* Jordan–Pólya number * Kempner function * Lah number * Lanczos approximation * Lozanić's triangle * Macaulay representation of an integer * Mahler's theorem *

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 die rolled ''n'' times. For ''n'' statistical independence, indepen ...

Multinomial coefficient 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 multinomials. Theorem For any positive integer ...

Multinomial formula 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 multinomials. Theorem For any positive integer a ...

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 multinomials. Theorem For any positive integer ...

* Multiplicities of entries in Pascal's triangle *

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 each element is called the ''multiplicity'' of ...

Multivariate gamma function In mathematics, the multivariate gamma function Γ''p'' is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the m ...

Narayana numbers Narayana (, ) is one of the forms and epithets of Vishnu. In this form, the deity is depicted in yogic slumber under the Kshira Sagara, celestial waters, symbolising the masculine principle and associated with his role of creation. He is als ...

Negative binomial distribution In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...

* Nörlund–Rice integral * Pascal matrix *

Pascal's pyramid In mathematics, Pascal's pyramid is a three-dimensional arrangement of the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which ...

Pascal's simplex In mathematics, Pascal's pyramid is a three-dimensional arrangement of the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which c ...

Permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

** List of permutation topics *

Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end ...

(also falling, lower, rising, upper factorials) *

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 if these events occur with a known const ...

Polygamma function In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) ...

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 multiplying positive integers, the function ...

* Proof of Bertrand's postulate * Sierpinski triangle * Star of David theorem *

Stirling number In mathematics, Stirling numbers arise in a variety of Analysis (mathematics), analytic and combinatorics, combinatorial problems. They are named after James Stirling (mathematician), James Stirling, who introduced them in a purely algebraic setti ...

Stirling transform In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by :b_n=\sum_^n \left\ a_k, where \left\ is the Stirling number of the second kind, which is the number of partitions of a set of size n into k ...

Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...

* Subfactorial *

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 :f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n where : is the binomial coefficient and (s)_n is the rising factorial, fall ...

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

* Trinomial expansion * Vandermonde's identity *

Wilson prime In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century ...

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''. That is (using the notations of ...

Wolstenholme prime In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes ...

Factorial and binomial topics In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...