{{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> factorials. They are a special case of the Jordan–Pólya numbers, 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}\\ &=\frac{(n!)^{n+1}}{\prod_{i=1}^{n} i^{i}} = \frac{(n!)^{n+1}}{H(n)} \end{align}</math>where <math>H</math> is the hyperfactorial.
Following the usual convention for the empty product, the superfactorial of 0 is 1. The 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 continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as <math>sf(n) = G(n+2)</math> for all nonnegative integers.{{r|barnes}}
According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when <math>p</math> is an 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== <references>
<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 | 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>
</references>
==External links== *{{MathWorld|id=Superfactorial|title=Superfactorial|mode=cs2}}
Category:Integer sequences Category:Factorial and binomial topics