Editing Superfactorial

{{Short description|Product of consecutive factorial numbers}}
{{Use dmy dates|cs1-dates=ly|date=December 2021}}
{{Use list-defined references|date=December 2021}}
In [[mathematics]], and more specifically [[number theory]], the '''superfactorial''' of a positive [[integer]] <math>n</math> is the product of the first <math>n</math> [[factorial]]s. They are a special case of the [[Jordan–Pólya number]]s, which are products of arbitrary collections of factorials.

==Definition==
The <math>n</math>th superfactorial <math>\mathit{sf}(n)</math> may be defined as:{{r|oeis}}
<math display=block>\begin{align}
\mathit{sf}(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_{i=1}^{n} i! = n!\cdot\mathit{sf}(n-1)\\
&= 1^n \cdot 2^{n-1} \cdot \cdots n = \prod_{i=1}^{n} i^{n+1-i}.\\
\end{align}</math>
Following the usual convention for the [[empty product]], the superfactorial of 0 is 1. The [[integer sequence|sequence]] of superfactorials, beginning with <math>\mathit{sf}(0)=1</math>, is:{{r|oeis}}
{{bi|left=1.6|1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... {{OEIS|A000178}}}}

==Properties==
Just as the factorials can be [[continuous function|continuously]] interpolated by the [[gamma function]], the superfactorials can be continuously interpolated by the [[Barnes G-function]].{{r|barnes}}

According to an analogue of [[Wilson's theorem]] on the behavior of factorials [[modular arithmetic|modulo]] [[prime number|prime]] numbers, when <math>p</math> is an [[parity (mathematics)|odd]] prime number
<math display=block>\mathit{sf}(p-1)\equiv(p-1)!!\pmod{p},</math>
where <math>!!</math> is the notation for the [[double factorial]].{{r|wilson}}

For every integer <math>k</math>, the number <math>\mathit{sf}(4k)/(2k)!</math> is a [[square number]]. This may be expressed as stating that, in the formula for <math>\mathit{sf}(4k)</math> as a product of factorials, omitting one of the factorials (the middle one, <math>(2k)!</math>) results in a square product.{{r|square}} Additionally, if any <math>n+1</math> integers are given, the product of their pairwise differences is always a multiple of <math>\mathit{sf}(n)</math>, and equals the superfactorial when the given numbers are consecutive.{{r|oeis}}

==References==
{{reflist|refs=

<ref name=barnes>{{citation
 | 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>

<ref name=oeis>{{cite OEIS|1=A000178 |2=Superfactorials: product of first n factorials|mode=cs2}}</ref>

<ref name=square>{{citation
 | last1 = White | first1 = D.
 | last2 = Anderson | first2 = M.
 | date = October 2020
 | doi = 10.1080/10511970.2020.1809039
 | issue = 10
 | journal = [[PRIMUS (journal)|PRIMUS]]
 | pages = 1038–1051
 | title = Using a superfactorial problem to provide extended problem-solving experiences
 | volume = 31| s2cid = 225372700
 }}</ref>

<ref name=wilson>{{citation
 | 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>

}}

==External links==
*{{MathWorld|id=Superfactorial|title=Superfactorial|mode=cs2}}

[[Category:Integer sequences]]
[[Category:Factorial and binomial topics]]