Editing Factorial (section)

==Related sequences and functions==
{{main|List of factorial and binomial topics}}
Several other integer sequences are similar to or related to the factorials:

;Alternating factorial
:The [[alternating factorial]] is the absolute value of the [[alternating sum]] of the first <math>n</math> factorials, {{nowrap|<math display=inline>\sum_{i = 1}^n (-1)^{n - i}i!</math>.}} These have mainly been studied in connection with their primality; only finitely many of them can be prime, but a complete list of primes of this form is not known.<ref>{{harvnb|Guy|2004}}. "B43: Alternating sums of factorials". pp. 152–153.</ref>
;Bhargava factorial
:The [[Bhargava factorial]]s are a family of integer sequences defined by [[Manjul Bhargava]] with similar number-theoretic properties to the factorials, including the factorials themselves as a special case.<ref name=bhargava/>
;Double factorial
:The product of all the odd integers up to some odd positive {{nowrap|integer <math>n</math>}} is called the [[double factorial]] {{nowrap|of <math>n</math>,}} and denoted by {{nowrap|<math>n!!</math>.<ref name="callan">{{cite arXiv|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|eprint=0906.1317|year=2009|class=math.CO}}</ref>}} That is, <math display=block>(2k-1)!! = \prod_{i=1}^k (2i-1) = \frac{(2k)!}{2^k k!}.</math> For example, {{nowrap|1=9!! = 1 × 3 × 5  × 7 × 9 = 945}}. Double factorials are used in [[List of integrals of trigonometric functions|trigonometric integrals]],<ref>{{cite journal
 | last = Meserve | first = B. E.
 | doi = 10.2307/2306136
 | issue = 7
 | journal = [[The American Mathematical Monthly]]
 | mr = 1527019
 | pages = 425–426
 | title = Classroom Notes: Double Factorials
 | volume = 55
 | year = 1948| jstor = 2306136
 }}</ref> in expressions for the [[gamma function]] at [[half-integer]]s and the [[Volume of an n-ball|volumes of hyperspheres]],<ref>{{cite journal|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8|s2cid=120103389}}.</ref> and in counting [[rooted binary tree|binary trees]] and [[perfect matching]]s.<ref name="callan"/><ref>{{cite journal
 | last1 = Dale | first1 = M. R. T.
 | last2 = Moon | first2 = J. W.
 | doi = 10.1016/0378-3758(93)90035-5
 | issue = 1
 | journal = [[Journal of Statistical Planning and Inference]]
 | mr = 1209991
 | pages = 75–87
 | title = The permuted analogues of three Catalan sets
 | volume = 34
 | year = 1993}}.</ref>
;Exponential factorial
:Just as [[triangular number]]s sum the numbers from <math>1</math> {{nowrap|to <math>n</math>,}} and factorials take their product, the [[exponential factorial]] exponentiates. The exponential factorial is defined recursively {{nowrap|as <math>a_0 = 1,\ a_n = n^{a_{n - 1}}</math>.}} For example, the exponential factorial of 4 is <math display=block>4^{3^{2^{1}}}=262144.</math> These numbers grow much more quickly than regular factorials.<ref>{{cite journal | last1 = Luca | first1 = Florian | author1-link = Florian Luca | last2 = Marques | first2 = Diego | issue = 3 | journal = [[Journal de Théorie des Nombres de Bordeaux]] | mr = 2769339 | pages = 703–718 | title = Perfect powers in the summatory function of the power tower | url = http://jtnb.cedram.org/item?id=JTNB_2010__22_3_703_0 | volume = 22 | year = 2010| doi = 10.5802/jtnb.740 | doi-access = free }}</ref>
;Falling factorial
:The notations <math>(x)_{n}</math> or <math>x^{\underline n}</math> are sometimes used to represent the product of the greatest <math>n</math> integers counting up to and {{nowrap|including <math>x</math>,}} equal to {{nowrap|<math>x!/(x-n)!</math>.}} This is also known as a [[Falling and rising factorials|falling factorial]] or backward factorial, and the <math>(x)_{n}</math> notation is a Pochhammer symbol.{{sfn|Graham|Knuth|Patashnik|1988|pp=x, 47–48}} Falling factorials count the number of different sequences of <math>n</math> distinct items that can be drawn from a universe of <math>x</math> items.<ref>{{cite book | last = Sagan | first = Bruce E. | author-link = Bruce Sagan | contribution = Theorem 1.2.1 | contribution-url = https://books.google.com/books?id=DYgEEAAAQBAJ&pg=PA5 | isbn = 978-1-4704-6032-7 | location = Providence, Rhode Island | mr = 4249619 | page = 5 | publisher = American Mathematical Society | series = Graduate Studies in Mathematics | title = Combinatorics: the Art of Counting | volume = 210 | year = 2020}}</ref> They occur as coefficients in the [[higher derivative]]s of polynomials,<ref>{{cite book|first=G. H.|last=Hardy|author-link=G. H. Hardy|title=A Course of Pure Mathematics|title-link=A Course of Pure Mathematics|edition=3rd|publisher=Cambridge University Press|year=1921|contribution=Examples XLV|page=215|contribution-url=https://archive.org/details/coursepuremath00hardrich/page/n229}}</ref> and in the [[factorial moment]]s of [[random variable]]s.<ref>{{cite book | last1 = Daley | first1 = D. J. | last2 = Vere-Jones | first2 = D. | contribution = 5.2: Factorial moments, cumulants, and generating function relations for discrete distributions | contribution-url = https://books.google.com/books?id=Af7lBwAAQBAJ&pg=PA112 | isbn = 0-387-96666-8 | location = New York | mr = 950166 | page = 112 | publisher = Springer-Verlag | series = Springer Series in Statistics | title = An Introduction to the Theory of Point Processes | year = 1988}}</ref>
;Hyperfactorials
:The [[hyperfactorial]] of <math>n</math> is the product <math>1^1\cdot 2^2\cdots n^n</math>. These numbers form the [[discriminant]]s of [[Hermite polynomials]].<ref>{{cite OEIS | 1=A002109 | 2=Hyperfactorials: Product_{k = 1..n} k^k}}</ref> They can be continuously interpolated by the [[K-function]],<ref>{{cite journal | last = Kinkelin | first = H. | author-link = Hermann Kinkelin | doi = 10.1515/crll.1860.57.122 | journal = [[Crelle's Journal | Journal für die reine und angewandte Mathematik]] | language = de | pages = 122–138 | title = Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung | trans-title = On a transcendental variation of the gamma function and its application to the integral calculus | volume = 1860 | year = 1860| issue = 57 | s2cid = 120627417 }}</ref> and obey analogues to Stirling's formula<ref>{{cite journal | last = Glaisher | first = J. W. L. | author-link = James Whitbread Lee Glaisher | journal = [[Messenger of Mathematics]] | pages = 43–47 | title = On the product {{math|1<sup>1</sup>.2<sup>2</sup>.3<sup>3</sup>...''n''<sup>''n''</sup>}} | url = https://archive.org/details/messengermathem01glaigoog/page/n56 | volume = 7 | year = 1877}}</ref> and Wilson's theorem.<ref>{{cite journal | last1 = Aebi | first1 = Christian | last2 = Cairns | first2 = Grant | doi = 10.4169/amer.math.monthly.122.5.433 | issue = 5 | journal = [[The American Mathematical Monthly]] | jstor = 10.4169/amer.math.monthly.122.5.433 | mr = 3352802 | pages = 433–443 | title = Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials | volume = 122 | year = 2015| s2cid = 207521192 }}</ref>
;Jordan–Pólya numbers
:The [[Jordan–Pólya number]]s are the products of factorials, allowing repetitions. Every [[Tree (graph theory)|tree]] has a [[symmetry group]] whose number of symmetries is a Jordan–Pólya number, and every Jordan–Pólya number counts the symmetries of some tree.<ref>{{cite OEIS|A001013|Jordan-Polya numbers: products of factorial numbers}}</ref>
;Primorial
:The [[primorial]] <math>n\#</math> is the product of [[prime number]]s less than or equal {{nowrap|to <math>n</math>;}} this construction gives them some similar divisibility properties to factorials,<ref name=caldwell-gallot/> but unlike factorials they are [[squarefree]].<ref>{{cite book | last = Nelson | first = Randolph | doi = 10.1007/978-3-030-37861-5 | isbn = 978-3-030-37861-5 | location = Cham | mr = 4297795 | page = 127 | publisher = Springer | title = A Brief Journey in Discrete Mathematics | url = https://books.google.com/books?id=m8PPDwAAQBAJ&pg=PA127 | year = 2020| s2cid = 213895324 }}</ref> As with the [[factorial prime]]s {{nowrap|<math>n!\pm 1</math>,}} researchers have studied [[primorial prime]]s {{nowrap|<math>n\#\pm 1</math>.<ref name=caldwell-gallot/>}}
;Subfactorial
:The [[subfactorial]] yields the number of [[derangement]]s of a set of <math>n</math> objects. It is sometimes denoted <math>!n</math>, and equals the closest integer {{nowrap|to <math>n!/e</math>.{{sfn|Graham|Knuth|Patashnik|1988|p=195}}}}
;Superfactorial
:The [[superfactorial]] of <math>n</math> is the product of the first <math>n</math> factorials. The superfactorials are continuously interpolated by the [[Barnes G-function]].<ref>{{cite journal|last=Barnes|first=E. W.|author-link=Ernest Barnes|jfm=30.0389.02|journal=[[The Quarterly Journal of Pure and Applied Mathematics]]|pages=264–314|title=The theory of the {{mvar|G}}-function|url=https://gdz.sub.uni-goettingen.de/id/PPN600494829_0031?tify={%22pages%22:[268],%22view%22:%22toc%22}|volume=31|year=1900}}</ref>