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Product of numbers from 1 to n
In mathematics, the factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers
Factorial
Mathematical function
In mathematics, the double factorial, or semifactorial, n‼ of a positive integer n is the product of all the positive integers up to n that have the same
Double_factorial
Topics referred to by the same term
{\displaystyle n} . Factorial may also refer to: Factorial experiment, a statistical experiment over all combinations of values Factorial code, data representation
Factorial_(disambiguation)
Mathematical functions
mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
Falling_and_rising_factorials
Extension of the factorial function
} , capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma
Gamma_function
Numeral 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
Factorial_number_system
Recursive mathematical formula
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 in a right-grouping
Exponential_factorial
Statistical experimental design approach
statistics, a fractional factorial design is a way to conduct experiments with fewer experimental runs than a full factorial design. Instead of testing
Fractional_factorial_design
Experimental design in statistics
In statistics, a factorial experiment (also known as full factorial experiment) investigates how multiple factors influence a specific outcome, called
Factorial_experiment
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
Alternating_factorial
Data representation for machine learning
in the data vector. When this occurs, it can be desirable to create a factorial code of the data, i.e., a new vector-valued representation of each data
Factorial_code
Expectation or average of the falling factorial of a random variable
the factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. Factorial moments
Factorial_moment
Collection of statistical models
accepted by the emerging field of psychology which developed strong (full factorial) experimental methods to which randomization and blinding were soon added
Analysis_of_variance
Generalization of the mathematical factorial
In mathematics, Bhargava's factorial function, or simply the Bhargava factorial, is a generalization of the factorial function developed by the Fields
Bhargava_factorial
Features in Haskell programming language
the factorial function for non-negative integers, shown in Haskell: factorial :: Integer -> Integer factorial 0 = 1 factorial n = n * factorial (n-1)
Haskell_features
Recursive integer sequence
triangle Catalan–Mersenne number Delannoy number Fuss–Catalan number List of factorial and binomial topics Lobb numbers Motzkin number Narayana number Narayana
Catalan_number
In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation
Factorial_moment_measure
Software programming optimization technique
the factorial of n: function factorial (n is a non-negative integer) if n is 0 then return 1 [by the convention that 0! = 1] else return factorial(n –
Memoization
Use of functions that call themselves
condition that terminates the recursion. An example is computing the factorial of an integer n, which is the product of all integers from 0 to n. For
Recursion_(computer_science)
Statistical method
acceptable mathematically. But different factorial theories proved to differ as much in terms of the orientations of factorial axes for a given solution as in
Factor_analysis
Subroutine call performed as final action of a procedure
factorial of the factorial: function factorial(n::Integer)::Integer if n == 0 return 1 else return n * factorial(n - 1) end end Indeed, n * factorial(n
Tail_call
Mathematical operation with only one operand
notation (e.g. negation ¬, additive inverse −), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g
Unary_operation
Prime number one less or more than a factorial
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
Factorial_prime
Zero after the final non-zero digit of a number
Number of trailing zeros for any factorial Python program to calculate the number of trailing zeros for any factorial Archived 2017-02-22 at the Wayback
Trailing_zero
Functional programming language
Refal ("Recursive functions algorithmic language"; Russian: РЕФАЛ) "is a functional programming language oriented toward symbolic computations", including
Refal
Approximation for factorials
approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small
Stirling's_approximation
Feature of some compilers
Factorial { enum { VALUE = N * Factorial<N - 1>::VALUE }; }; template <> struct Factorial<0> { enum { VALUE = 1 }; }; void foo() { int x = Factorial<0>::VALUE;
Compile-time function execution
Compile-time_function_execution
a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem Alternating factorial Antichain Beta
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Natural number
smallest composite Wagstaff number with prime index 185,794,560 = double factorial of 18 188,378,402 = number of ways to partition {1,2,...,11} and then
100,000,000
Object-oriented programming language
chained by writing them one after another: 3 factorial factorial log which sends "factorial" to 3, then "factorial" to the result (6), then "log" to the result
Smalltalk
Programming style in which control is passed explicitly
global transformation. The direct-style factorial takes, as might be expected, a single argument; the CPS factorial& takes two: the argument and a continuation
Continuation-passing_style
Statically typed functional programming language
optimization: pub fn factorial(x: Int) -> Int { // The public function calls the private tail recursive function factorial_loop(x, 1) } fn factorial_loop(x: Int
Gleam_(programming_language)
Functional programming language
matching) factorial 0 = 1 factorial n = n * factorial (n - 1) -- Using recursion (with guards) factorial n | n < 2 = 1 | otherwise = n * factorial (n - 1)
Haskell
Experimental designs for response surface methodology
proficient and more powerful than other designs such as the three-level full factorial design, central composite design (CCD) and Doehlert design, despite its
Box–Behnken_design
Statistical phenomenon where some effects appear the same
In the statistical theory of factorial experiments, aliasing is the property of fractional factorial designs that makes some effects "aliased" with each
Aliasing (factorial experiments)
Aliasing_(factorial_experiments)
Mathematical series, portmanteau of "Fibonacci" and "factorial"
In mathematics, the Fibonorial n!F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive
Fibonorial
Study of programming languages via mathematical objects
F(Map<int,int> factorial_less_defined) { Map<int,int> new_factorial = Map.empty(); for (int n in all<int>()) { if (f = factorial_nonrecursive(factorial_less_defined
Denotational_semantics
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable
Factorial moment generating function
Factorial_moment_generating_function
Canadian-American mathematician (born 1974)
Hanke) of the 290 theorem. A novel generalization of the factorial function, Bhargava factorial, providing an answer to a decades-old question of George
Manjul_Bhargava
TXL is a special-purpose programming language originally designed by Charles Halpern-Hamu and James Cordy at the University of Toronto in 1985. The acronym
TXL_(programming_language)
Number
sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product. The role of
0
Type of permutation of a set of elements
subfactorial Dn equals the nearest integer to n!/e, where n! denotes the factorial of n and e ≈ 2.718281828... is Euler's number. The problem of counting
Derangement
Type of integral domain
mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement
Unique_factorization_domain
Concept in combinatorics (part of mathematics)
of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product ( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a )
Q-Pochhammer_symbol
In the statistical analysis of the results from factorial experiments, the sparsity-of-effects principle states that a system is usually dominated by
Sparsity-of-effects_principle
Punctuation mark (!)
text without emotion. Other uses include: In mathematics, it denotes the factorial operation and shriek maps. Several computer languages use ! at the beginning
Exclamation_mark
General-purpose functional programming language
used for abstraction. The factorial function can be expressed as follows: fun factorial n = if n = 0 then 1 else n * factorial (n - 1) An SML compiler must
Standard_ML
more than one factor. It is thus a non-parametric alternative to multi-factorial ANOVA analyses. The test is named after James Scheirer, William Ray and
Scheirer–Ray–Hare_test
Vehicle propelled fully or mostly by electricity
challenges. Recent commercial developments include Mercedes-Benz and Factorial Energy conducting road tests of semi-solid-state batteries in the EQS
Electric_vehicle
Product of the first "n" prime numbers
is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers,
Primorial
Statistical property of measurement
equation models, including CFA, measurement invariance is often termed factorial invariance. In the common factor model, measurement invariance may be
Measurement_invariance
Natural number
prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a factorial prime, a Harshad number, a lucky prime, a happy number (happy prime),
7
General-purpose programming language
ValueError('You must enter a non-negative integer') factorial = 1 for i in range(2, n + 1): factorial *= i print(factorial) Python's large standard library is commonly
Python_(programming_language)
Natural number
so that 1 is also the identity for any power semigroup. 1 is its own factorial 1 ! = 1 {\displaystyle 1!=1} . Moreover, the empty product, that is the
1
Experimental design in statistical mathematics
for the response variable without needing to use a complete three-level factorial experiment. After the designed experiment is performed, linear regression
Central_composite_design
Number theory expression
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
Legendre's_formula
In mathematics, the n! conjecture is the conjecture that the dimension of a certain bi-graded module of diagonal harmonics is n!. It was made by A. M.
N!_conjecture
Factorial method
Multiple factor analysis (MFA) is a factorial method devoted to the study of tables in which a group of individuals is described by a set of variables
Multiple_factor_analysis
Recursive function
following: factorial 5 = 5 * (factorial 4) = 120 factorial 4 = 4 * (factorial 3) = 24 factorial 3 = 3 * (factorial 2) = 6 factorial 2 = 2 * (factorial 1) =
Hylomorphism (computer science)
Hylomorphism_(computer_science)
Metaprogramming technique
factorial(uint32_t n) { return n == 0 ? 1 : n * factorial(n - 1); } // Usage examples: // factorial(0) would yield 1; // factorial(4) would yield 24. The code above
Template_metaprogramming
Branch of discrete mathematics
identities Factorials & approximations Factorial · Bhargava factorial · Hyperfactorial · Alternating factorial · Factorial moment · Factorial number system
Combinatorics
Software that is used for designing factorial experiments plays an important role in scientific experiments and represents a route to the implementation
Multifactor design of experiments software
Multifactor_design_of_experiments_software
Family of mathematical integrals
Expanding W 2 p {\displaystyle W_{2p}} and using the formula above for the factorials, we get W 2 p = ( 2 p ) ! 2 2 p ( p ! ) 2 ⋅ π 2 ∼ C ( 2 p e ) 2 p 2 p
Wallis'_integrals
Polynomial sequence
k}E_{k}(x).} The Bernoulli polynomials may be expanded in terms of the falling factorial ( x ) k {\displaystyle (x)_{k}} as B n + 1 ( x ) = B n + 1 + ∑ k = 0 n
Bernoulli_polynomials
Topics referred to by the same term
for the same memory location Aliasing (factorial experiments), a property that makes some effects in factorial experiments "aliased", or indistinguishable
Aliasing_(disambiguation)
Series of economic reforms in Pakistan
The National Finance Commission Award (NFC) is a series of planned economic programs in Pakistan enacted since 1951. Constituted under the Article 160
National Finance Commission Award
National_Finance_Commission_Award
High-level computer programming language
calculate a factorial. % Accepts a number and calculates its factorial function factorial (n: int) : real if n = 0 then result 1 else result n * factorial (n -
Turing_(programming_language)
Mathematical constant
In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in various
Natural_logarithm_of_2
Function that interpolates the factorial
interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive
Pseudogamma_function
French mathematician
– 13 May 1826) was a French mathematician, who worked primarily with factorials. Christian Kramp's father was his teacher at grammar school in Strasbourg
Christian_Kramp
Number of subsets of a given size
\cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}},} which using factorial notation can be compactly expressed as ( n k ) = n ! k ! ( n − k ) !
Binomial_coefficient
Natural number
thirteen digits 311,809,494,089 = Markov prime 316,234,143,225 = double factorial of 23 321,253,732,800 = superior highly composite number 333,333,333,333
100,000,000,000
Sum of a factorial number and a triangular number
integer formed by adding a factorial and a triangular number with the same index. The name is a portmanteau of "factorial" and "triangular." For n ≥ 1
Factoriangular_number
Calculations where numbers' precision is only limited by computer memory
the successive factorial numbers. constants: Limit = 1000 % Sufficient digits. Base = 10 % The base of the simulated arithmetic. FactorialLimit = 365 %
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Estimate of number of possible chess games
6331 (8!)−2 (where the ! represents the factorial and the underlined superscript represents a falling factorial), or roughly 3.7×1034. This includes some
Shannon_number
Programming language
Nial (from "Nested Interactive Array Language") is a high-level array programming language developed from about 1981 by Mike Jenkins of Queen's University
Nial
Natural number
may be swapped but turning over is not allowed 13,749,310,575 = double factorial of 21 13,800,000,000 = approximate age of the universe in years 13,893
10,000,000,000
Cross-platform reverse-Polish calculator program
by defining a macro which (conditionally) reinvokes itself. A simple factorial of the top of the stack might be implemented as: # F(x): return x! # if
Dc_(computer_program)
Number-theoretic concept
integer n ≥ 0 {\displaystyle n\geq 0} has a unique representation in the factorial number system as n = ∑ i = 1 ∞ c i i ! with c i ∈ Z , {\displaystyle
Profinite_integer
Discrete probability distribution
binomial variation, and the two models have equal variances. The r-th factorial moment of a Beta-binomial random variable X is E [ ( X ) r ] = n ! (
Beta-binomial_distribution
Extension of the factorial function
gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance
Hadamard's_gamma_function
Computer programming concept
The factorial function is only defined where its parameter is an integer greater than or equal to zero. So an implementation of the factorial function
Precondition
Mathematical sequences in combinatorics
that many use for falling factorials is used in special functions for rising factorials.) Similarly, the rising factorial, defined as x ( n ) =
Stirling_number
Hebrew book on Jewish mysticism
one for the earliest known descriptions of the mathematical function factorial, enumerating its first seven values and emphasizing its rapid growth.
Sefer_Yetzirah
Multi-paradigm system programming language
function above: template Factorial(ulong n) { static if (n < 2) { enum Factorial = 1; } else { enum Factorial = n * Factorial!(n-1); } } In the following
D_(programming_language)
his/her hands. Robust parameter designs are very similar to fractional factorial designs (FFDs) in that the optimal design can be found using Hadamard
Robust_parameter_design
conditions that are both strictly stronger than the BFD condition are the half-factorial domain condition (HFD: any two factorizations of any given x have the
Atomic_domain
the context of factorial designs and regression models to distinguish main effects from interaction effects. Relative to a factorial design, under an
Main_effect
Programming language
factorial, are most naturally represented in a purely functional form. The following recursive, purely functional Caml function implements factorial:
Caml
function for computing the factorial of a number, written in E. Functions are defined using the def keyword. def factorial(n :int) :int { if (n == 1)
E_(programming_language)
Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied
Multiplicative partitions of factorials
Multiplicative_partitions_of_factorials
Proj of the symmetric algebra of V. Q-factorial A normal variety is Q {\displaystyle \mathbb {Q} } -factorial if every Q {\displaystyle \mathbb {Q} }
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Assessment tool for occupational burnout
properties of the MBI have proved to be problematic, for example, in terms of factorial validity (measuring a unitary construct) and measurement invariance, casting
Maslach_Burnout_Inventory
Natural number
which its factorial can be stored in IEEE 754 double-precision floating-point format. This is probably why it is also the largest factorial that Google's
170_(number)
Programming language family
notation. For example, to evaluate a number's factorial: (defun factorial (n) (if (zerop n) 1 (* n (factorial (1- n))))) An alternative implementation takes
Lisp_(programming_language)
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here
List_of_mathematical_series
Compare means Multiple comparison Designs Completely randomized Factorial Fractional factorial Plackett–Burman Taguchi Response surface methodology Polynomial
Fisher's_inequality
Higher-order function Y for which Y f = f (Y f)
<- \(f) { g <- \(x) f(x(x)) g(g) } This can then be used to implement factorial as follows: fact <- \(f) \(n) if (n == 0) 1 else n * f(n - 1) Y(fact)(5)
Fixed-point_combinator
In mathematics, when is n!+1 a square
{\displaystyle n!+1} is a perfect square, where n ! {\displaystyle n!} is the factorial. Only three values of n {\displaystyle n} are known — 4, 5, 7 — and it
Brocard's_problem
Mathematical version of an order change
RNA sequences. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers
Permutation
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