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,

*n*! is read as "

*n*factorial". The current notation was introduced by the mathematician Christian Kramp in 1808.

Table of contents |

2 Generalization to the Gamma function 3 Multifactorials 4 Hyperfactorials 5 Superfactorials 6 External link |

## Introduction

Factorials are often used as a simple example when teaching recursion in computer science because they satisfy the following recursive relationship (if*n*≥ 1):

*n*! =*n*(*n*− 1)!

- 0! = 1

- 0! is an instance of the empty product, and therefore 1
- it makes the above recursive relation work for
*n*= 1 - many identities in combinatorics would not work for zero sizes without this definition

*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

*x*

^{n}is

*n*!.

When *n* is large, *n*! can be estimated quite accurately using Stirling's approximation:

## Generalization to the Gamma function

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 by the property:

*n*is any non-negative integer.

## Multifactorials

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, (2*n*)!! = 2^{n}*n*! and (2*n*+1)! = (2*n*+1)!! 2^{n}*n*!. The double factorial is related to the Gamma function of half-integer order by Γ(*n*+1/2) = √π (2*n*-1)!!/2^{n}.

One should be careful not to interpret *n*!! as the factorial of *n*!, 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.

## Hyperfactorials

Occasionally the**hyperfactorial**of

*n*is considered. It is written as

*H*(

*n*) and defined by

*H*(*n*) =*n*^{n}(*n*-1)^{(n-1)}... 3^{3}2^{2}1^{1}

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.

## Superfactorials

The**superfactorial**of

*n*, written as

*n*$ (a factorial sign with an S written over it) has been defined as

*n*$ =*n*!^{(4)}*n*!

^{(4)}notation denotes the hyper4 operator, or using Knuth's up-arrow notation,