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In mathematics, the factorial of a positive integer n, denoted n!, is the product of the positive integers less than or equal to n. For example,
Usually, n! is read as "n factorial". The current notation was introduced by the mathematician Christian Kramp in 1808.
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2 Generalization to the Gamma function 3 Multifactorials 4 Hyperfactorials 5 Superfactorials 6 External link |
Factorials are often used as a simple example when teaching recursion in computer science because they satisfy the following recursive relationship (if n ≥ 1):
Introduction
In addition, one defines
for several related reasons:
Factorials are important in combinatorics because there are n! different ways of arranging n distinct objects in a sequence (see permutation). They also turn up in formulas of calculus, such as in Taylor's theorem, for instance, because
the n-th derivative of the function xn is n!.
When n is large, n! can be estimated quite accurately using Stirling's approximation:
The related Gamma function Γ(z) is defined for all complex numbers z except for z = 0, -1, -2, -3, ... It is related to the factorial in that it satisfies the same (above-mentioned) recursive relationship as the factiorial function (offset by one):
Because of this relationship, the gamma function is often thought of as a generalization of the factorial function to the domain of complex numbers. This is legitimate because:
A common related notation is to use multiple exclamation points (!) to denote a multifactorial, the product of integers in steps of two, three, or more.
For example, n!! denotes the double factorial of n, defined recursively by n!! = n (n-2)!! for n > 1 and as 1 for n = 0,1. Thus, (2n)!! = 2nn! and (2n+1)! = (2n+1)!! 2nn!. The double factorial is related to the Gamma function of half-integer order by Γ(n+1/2) = √π (2n-1)!!/2n.
One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number.
The double factorial is the most commonly used variant, but one can similarly define the triple factorial (!!!) and so on. In general, the k-th factorial, denoted by !(k), is defined recursively by: n!(k) = n (n-k)!(k) for n > k-1, n!(k) = n for k > n > 0, and 0!(k) = 1.
Occasionally the hyperfactorial of n is considered. It is written as H(n)
and defined by
The hyperfactorial function is similar to the factorial, but produces larger numbers. The rate of growth of this function, however, is not much larger than a regular factorial.
The superfactorial of n, written as n$ (a factorial sign with an S written over it) has been defined as
Generalization to the Gamma function
Consequently, the gamma and factorial functions equate (with an offset)
when n is any non-negative integer.Multifactorials
Hyperfactorials
E.g. H(4) = 27648.Superfactorials
where the (4) notation denotes the hyper4 operator, or using Knuth's up-arrow notation,
External link