Product of consecutive factorial numbers
In
mathematics , and more specifically
number theory , the superfactorial of a positive
integer
n
{\displaystyle n}
is the product of the first
n
{\displaystyle n}
factorials . They are a special case of the
Jordan–Pólya numbers , which are products of arbitrary collections of factorials.
Definition
The
n
{\displaystyle n}
th superfactorial
s
f
(
n
)
{\displaystyle {\mathit {sf}}(n)}
may be defined as:
[1]
s
f
(
n
)
=
1
!
⋅
2
!
⋅
⋯
n
!
=
∏
i
=
1
n
i
!
=
n
!
⋅
s
f
(
n
−
1
)
=
1
n
⋅
2
n
−
1
⋅
⋯
n
=
∏
i
=
1
n
i
n
+
1
−
i
.
{\displaystyle {\begin{aligned}{\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{aligned}}}
Following the usual convention for the
empty product , the superfactorial of 0 is 1. The
sequence of superfactorials, beginning with
s
f
(
0
)
=
1
{\displaystyle {\mathit {sf}}(0)=1}
, is:
[1]
1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (sequence
A000178 in the
OEIS )
Properties
Just as the factorials can be
continuously interpolated by the
gamma function , the superfactorials can be continuously interpolated by the
Barnes G-function .
[2]
According to an analogue of
Wilson's theorem on the behavior of factorials
modulo
prime numbers, when
p
{\displaystyle p}
is an
odd prime number
s
f
(
p
−
1
)
≡
(
p
−
1
)
!
!
(
mod
p
)
,
{\displaystyle {\mathit {sf}}(p-1)\equiv (p-1)!!{\pmod {p}},}
where
!
!
{\displaystyle !!}
is the notation for the
double factorial .
[3]
For every integer
k
{\displaystyle k}
, the number
s
f
(
4
k
)
/
(
2
k
)
!
{\displaystyle {\mathit {sf}}(4k)/(2k)!}
is a
square number . This may be expressed as stating that, in the formula for
s
f
(
4
k
)
{\displaystyle {\mathit {sf}}(4k)}
as a product of factorials, omitting one of the factorials (the middle one,
(
2
k
)
!
{\displaystyle (2k)!}
) results in a square product.
[4] Additionally, if any
n
+
1
{\displaystyle n+1}
integers are given, the product of their pairwise differences is always a multiple of
s
f
(
n
)
{\displaystyle {\mathit {sf}}(n)}
, and equals the superfactorial when the given numbers are consecutive.
[1]
References
^
a
b
c
Sloane, N. J. A. (ed.),
"Sequence A000178 (Superfactorials: product of first n factorials)" , The
On-Line Encyclopedia of Integer Sequences , OEIS Foundation
^
Barnes, E. W. (1900),
"The theory of the G -function" ,
The Quarterly Journal of Pure and Applied Mathematics , 31 : 264–314,
JFM
30.0389.02
^ Aebi, Christian; Cairns, Grant (2015), "Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials",
The American Mathematical Monthly , 122 (5): 433–443,
doi :
10.4169/amer.math.monthly.122.5.433 ,
JSTOR
10.4169/amer.math.monthly.122.5.433 ,
MR
3352802 ,
S2CID
207521192
^ White, D.; Anderson, M. (October 2020), "Using a superfactorial problem to provide extended problem-solving experiences",
PRIMUS , 31 (10): 1038–1051,
doi :
10.1080/10511970.2020.1809039 ,
S2CID
225372700
External links