A268504 -id:A268504 - cp's OEIS frontend

A098694 Double-superfactorials: a(n) = Product_{k=1..n} (2k)!.

1, 2, 48, 34560, 1393459200, 5056584744960000, 2422112183371431936000000, 211155601241022491077587763200000000, 4417964278440225627098723475313498521600000000000

Offset: 0

Ralf Stephan, Sep 22 2004

nonn

Hankel transform of double factorial numbers A001147. - Paul Barry, Jan 28 2008

Hankel transform of A112934(n+1). - Paul Barry, Dec 04 2009

Seiichi Manyama, Table of n, a(n) for n = 0..28
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

Cf. A000178, A010050, A074962, A168467, A268504, A268505, A268506, A271946, A271947.

[&*[ Factorial(2*k): k in [0..n] ]: n in [0..10]]; // Vincenzo Librandi, Dec 11 2016

Table[Product[(2k)!,{k,1,n}],{n,0,10}] (* Vaclav Kotesovec, Nov 13 2014 *)

a(n) = prod(k=1, n, (2*k)!); \\ Michel Marcus, Dec 11 2016

from math import prod
def A098694(n): return prod(((k+1)*((k<<1)+1)<<1)**(n-k) for k in range(1,n+1))<Chai Wah Wu, Nov 26 2023

a(n) = Product_{k=0..n} (2*(k+1)*(2*k+1))^(n-k). - Paul Barry, Jan 28 2008
a(n) = A000178(n)*A057863(n)*A006125(n+1) = A121835(n)*A006125(n+1). - Paul Barry, Jan 28 2008
G.f.: G(0)/(2*x)-1/x, where G(k)= 1  + 1/(1 - 1/(1 + 1/(2*k+2)!/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) ~ 2^(n^2+2*n+17/24) * n^(n^2+3*n/2+11/24) * Pi^((n+1)/2) / (A^(1/2) * exp(3*n^2/2+3*n/2-1/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014
a(n) = A^(3/2)*2^(n^2+n-1/24)*Pi^(-n/2-1/4)*G(n+3/2)*G(n+2)/exp(1/8), where G(n) is the Barnes G-function and A is the Glaisher-Kinkelin constant. - Ilya Gutkovskiy, Dec 11 2016
a(n) = A000178(2*n + 1) / A168467(n). - Vaclav Kotesovec, Oct 28 2017
For n > 0, a(n) = 2^((n+1)/2) * n * sqrt(BarnesG(2*n)*Gamma(n)) * Gamma(2*n). - Vaclav Kotesovec, Nov 27 2024

A268506 a(n) = Product_{k=0..n} (5*k)!.

Original entry on oeis.org

1, 120, 435456000, 569434649591808000000, 1385378702517271000054360965120000000000, 21488900044302744250061179567064173417691432878080000000000000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

nonn

Partial products of A100734. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..16

Cf. A000178, A098694, A100734, A268504, A268505, A271946, A271947.

Mathematica
```
Table[Product[(5*k)!,{k,0,n}],{n,0,8}]
```

{a(n) = prod(k=1, n, (5*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ Gamma(1/5)^(3/5) * Gamma(2/5)^(2/5) * Gamma(3/5)^(1/5) * 2^(n/2 - 1/10) * Pi^(n/2 - 1/10) * 5^(23/60 + 3*n + 5*n^2/2) * exp(1/60 - 3*n - 15*n^2/4) * n^(41/60 + 3*n + 5*n^2/2) / A^(1/5), where A = A074962 is the Glaisher-Kinkelin constant.

A268196 a(n) = Product_{k=0..n} binomial(3*k,k).

Original entry on oeis.org

1, 3, 45, 3780, 1871100, 5618913300, 104309506501200, 12129109415959536000, 8920608231265175901456000, 41809329673499408044341517200000, 1256161937180234817183361549396758000000, 243113461110708695347467432844366521953760000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

Harvey P. Dale, Table of n, a(n) for n = 0..49
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A001142, A005809, A007685, A086205, A112332, A268504, A262261.

Table[Product[Binomial[3k,k],{k,0,n}],{n,0,12}]
FoldList[Times,Table[Binomial[3n,n],{n,0,15}]] (* Harvey P. Dale, Apr 23 2018 *)

a(n) = A^(7/6) * Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36)* BarnesG(n + 4/3) * BarnesG(n + 5/3) / (exp(7/72) * 2^(n^2 + 2*n + 5/8) * Pi^(n/2 + 5/12) * BarnesG(n + 3/2) * BarnesG(n + 2)), where A = A074962 is the Glaisher-Kinkelin constant.
a(n) ~ A^(7/6) * Gamma(1/3)^(1/3) * 3^(11/36 + 2*n + 3*n^2/2) * exp(n/2 - 7/72) / (2^(n^2 + 2*n + 7/8) * Pi^(n/2 + 2/3) * n^(n/2 + 25/72)), where A = A074962 is the Glaisher-Kinkelin constant.
a(n) = A268504(n) / (A000178(n) * A098694(n)).

A268505 a(n) = Product_{k=0..n} (4*k)!.

Original entry on oeis.org

1, 24, 967680, 463520268288000, 9698137182219213471744000000, 23594617426193665303453830729600860160000000000, 14639242671589099207353038379393488170313478620292159897600000000000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

nonn

Partial products of A100733. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..19

Cf. A000178, A098694, A100733, A268504, A268506, A271946, A271947.

Mathematica
```
Table[Product[(4*k)!,{k,0,n}],{n,0,8}]
```

{a(n) = prod(k=1, n, (4*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ Gamma(1/4)^(1/2) * 2^(5/6 + 11*n/2 + 4*n^2) * exp(1/48 - 5*n/2 - 3*n^2) * n^(29/48 + 5*n/2 + 2*n^2) * Pi^(1/4 + n/2) / A^(1/4), where A = A074962 is the Glaisher-Kinkelin constant.

a(n) = A^(15/4) * sqrt(Gamma(1/4)) * exp(-5/16) * 2^(4*n^2 + 7*n/2 - 5/12) * BarnesG(n + 5/4) * BarnesG(n + 3/2) * BarnesG(n + 7/4) * BarnesG(n+2) / Pi^(3*n/2 + 1). - Vaclav Kotesovec, Apr 23 2024

A100732 a(n) = (3*n)!.

Original entry on oeis.org

1, 6, 720, 362880, 479001600, 1307674368000, 6402373705728000, 51090942171709440000, 620448401733239439360000, 10888869450418352160768000000, 265252859812191058636308480000000

Offset: 0

Ralf Stephan, Dec 08 2004

a(n) equals (-1)^n times the determinant of the (3n+1) X (3n+1) matrix with consecutive integers from 1 to 3n+1 along the main diagonal, consecutive integers from 2 to 3n+1 along the superdiagonal, consecutive integers from 1 to 3n along the subdiagonal, and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 12 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000142, A010050, A100733, A100734, A143084, A143819, A268504.

a100732 = a000142 . a008585  -- Reinhard Zumkeller, Feb 19 2013

[Factorial(3*n): n in [0..15]]; // Vincenzo Librandi, Sep 24 2011

Table[(-1)^n*Det[Array[KroneckerDelta[#1, #2]*(#1 - 1) + KroneckerDelta[#1, #2 - 1]*(#1) + KroneckerDelta[#1, #2 + 1]*(#1 - 2) + 1 &, {3*n + 1, 3*n + 1}]], {n, 0, 24}] (* John M. Campbell, Jul 12 2011 *)
(3Range[0,10])! (* Harvey P. Dale, Sep 23 2011 *)

[factorial(3*n) for n in range(0, 11)] # Peter Luschny, Jun 06 2016

a(n) = A000142(A008585(n)).
E.g.f.: 1/(1-x^3).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(2*Pi)*3^(3*n+1/2)*n^(3*n+1/2)/exp(3*n).
Sum_{n>=0} 1/a(n) = (exp(3/2) + 2*cos(sqrt(3)/2))/(3*exp(1/2)) = A143819. (End)
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*exp(3/2)*cos(sqrt(3)/2))/(3*e). - Amiram Eldar, Feb 14 2021
a(n) = A143084(2n,n). - Alois P. Heinz, Jul 12 2024

A271946 a(n) = Product_{k=0..n} (6*k)!.

Original entry on oeis.org

1, 720, 344881152000, 2208058019165981638656000000, 1369986068925795885347091500568179543900160000000000, 363392722685428853076589064611759104109572860599125858715484081356800000000000000000

Offset: 0

Vaclav Kotesovec, Apr 17 2016

nonn

The next term has 126 digits.

Partial products of A195390. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..15

Cf. A000178, A098694, A195390, A268504, A268505, A268506, A271947.

Mathematica
```
Table[Product[(6*k)!,{k,0,n}],{n,0,8}]
```

{a(n) = prod(k=1, n, (6*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ A^(-1/6) * exp(1/72 - 7*n/2 - 9*n^2/2) * n^(55/72 + 7*n/2 + 3*n^2) * 2^(1/72 + 4*n + 3*n^2) * 3^(47/72 + 7*n/2 + 3*n^2) * Pi^(n/2 - 1/3) * Gamma(1/3)^(5/3), where A = A074962 is the Glaisher-Kinkelin constant.

A271947 a(n) = Product_{k=0..n} (7*k)!.

Original entry on oeis.org

1, 5040, 439378587648000, 22448266013011335649028997120000000, 6844214664110424043644485692109939233534721371668480000000000000

Offset: 0

Vaclav Kotesovec, Apr 17 2016

nonn

The next term has 104 digits.
In general, for p > 0, Product_{k=0..n} (p*k)! = p^((n+1)*(n*p+1)/2) * (2*Pi)^((n+1)*(1-p)/2) * Product_{j=1..p} BarnesG(n+1+j/p) / BarnesG(j/p).
Equivalently, Product_{k=0..n} (p*k)! = A^(p - 1/p) * exp(1/(12*p) - p/12) * (2*Pi)^((1-p)*n/2) * p^(p*n^2/2 + (p+1)*n/2 - 1/(12*p)) * Product_{j=1..p} (BarnesG(n + 1 + j/p) / Gamma(j/p)^(j/p)), where A = A074962 is the Glaisher-Kinkelin constant.
Asymptotics: Product_{k=0..n} (p*k)! ~ exp(p/12 - (p+1)*n/2 - 3*p*n^2/4) * n^(1/4 + p/12 + 1/(12*p) + (p+1)*n/2 + p*n^2/2) * p^(1/2 + (p+1)*n/2 + p*n^2/2) * (2*Pi)^(n/2 - p/4 + 3/4) / (A^p * Product_{j=1..p} BarnesG(j/p)).
Equivalently, Product_{k=0..n} (p*k)! ~ A^(-1/p) * exp(1/(12*p) - (p+1)*n/2 - 3*p*n^2/4) * n^(1/4 + p/12 + 1/(12*p) + (p+1)*n/2 + p*n^2/2) * p^(-1/(12*p) + (p+1)*n/2 + p*n^2/2) * (2*Pi)^(n/2 + (p+1)/4) / Product_{j=1..p-1} Gamma(j/p)^(j/p), where A = A074962 is the Glaisher-Kinkelin constant.
The general formula for the product of Barnes-G functions is: Product_{j=1..p} BarnesG(j/p) = A^(1/p - p) * exp(p/12 - 1/(12*p)) * p^(1/2 + 1/(12*p)) * (2*Pi)^((1-p)/2) * Product_{j=1..p-1} Gamma(j/p)^(j/p).

Seiichi Manyama, Table of n, a(n) for n = 0..13
Eric Weisstein's World of Mathematics, Barnes G-Function
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant

Cf. A000178 (p=1), A098694 (p=2), A268504 (p=3), A268505 (p=4), A268506 (p=5), A271946 (p=6).

Partial products of A195391.

Mathematica
```
Table[Product[(7*k)!,{k,0,n}],{n,0,8}]
```

{a(n) = prod(k=1, n, (7*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ A^(-1/7) * exp(1/84 - 4*n - 21*n^2/4) * n^(71/84 + 4*n + 7*n^2/2) * 7^(-1/84 + 4*n + 7*n^2/2) * (2*Pi)^(n/2 + 2) / (Gamma(1/7)^(1/7) * Gamma(2/7)^(2/7) * Gamma(3/7)^(3/7) * Gamma(4/7)^(4/7) * Gamma(5/7)^(5/7) * Gamma(6/7)^(6/7)), where A = A074962 is the Glaisher-Kinkelin constant.

A262261 a(n) = Product_{k=0..n} binomial(4*k,k).

Original entry on oeis.org

1, 4, 112, 24640, 44844800, 695273779200, 93581069585203200, 110803729631663996928000, 1165466869384731418887782400000, 109720873815210197693149787062272000000, 93006053830822450607559730484293052399616000000

Offset: 0

Vaclav Kotesovec, Apr 17 2016

In general, for p > 1, Product_{k=0..n} binomial(p*k,k) ~ A^(1 + 1/(p*(p-1))) * exp(n/2 - 1/12 - 1/(12*p*(p-1))) * n^(-1/3 - n/2 - 1/(12*p*(p-1))) * (p-1)^(1/(12*(p-1)) - p*n/2 - (p-1)*n^2/2) * p^(-1/(12*p) + (p+1)*n/2 + p*n^2/2) * (2*Pi)^(-1/4 - n/2) * Product_{j=1..p-1} (Gamma(j/(p-1))^(j/(p-1)) / Gamma(j/p)^(j/p)), where A = A074962 is the Glaisher-Kinkelin constant.

Cf. A007685 (p=2), A268196 (p=3).
Cf. A000178, A098694, A268504, A268505, A268506, A271946, A271947.
Cf. A005810, A165975.

Table[Product[Binomial[4*k,k],{k,0,n}],{n,0,10}]

a(n) ~ A^(13/12) * 2^(9*n/2 + 4*n^2) * exp(n/2 - 13/144) * Gamma(1/4)^(1/2) / (Gamma(1/3)^(1/3) * 3^(11/36 + 2*n + 3*n^2/2) * Pi^(7/12 + n/2) * n^(49/144 + n/2)), where A = A074962 is the Glaisher-Kinkelin constant.

A294319 a(n) = Product_{k=0..n} (3*k + 2)!.

Original entry on oeis.org

2, 240, 9676800, 386266890240000, 33674087438261157888000000, 11977449554394932435557703221248000000000, 29139961073721833036780987632259240162985246720000000000000

Offset: 0

Vaclav Kotesovec, Oct 28 2017

nonn

Cf. A268504, A294318.

Table[Product[(3*k + 2)!, {k, 0, n}] , {n, 0, 10}]
FoldList[Times,(3 Range[0,10]+2)!] (* Harvey P. Dale, Sep 26 2023 *)

a(n) ~ 3^(3*n^2/2 + 4*n + 101/36) * (2*Pi)^(n/2 + 1/3) * n^(3*n^2/2 + 4*n + 91/36) * Gamma(1/3)^(1/3) / (A^(1/3) * exp(9*n^2/4 + 4*n - 1/36)), where A is the Glaisher-Kinkelin constant A074962.

A268504(n) * A294318(n) * A294319(n) = A000178(3*n + 2).

A294318 a(n) = Product_{k=0..n} (3*k + 1)!.

Original entry on oeis.org

1, 24, 120960, 438939648000, 2733286318040678400000, 57187975336110258000180019200000000, 6956637001938940278070327452315517609574400000000000, 7819265053064003641840525064819521833578308036969094971392000000000000000

Offset: 0

Vaclav Kotesovec, Oct 28 2017

nonn

Cf. A168467, A268504, A294319.

Table[Product[(3*k + 1)!, {k, 0, n}] , {n, 0, 10}]

a(n) ~ 3^(3*n^2/2 + 3*n + 47/36) * n^(3*n^2/2 + 3*n + 49/36) * (2*Pi)^(n/2 + 5/6) / (A^(1/3) * Gamma(1/3)^(2/3) * exp(9*n^2/4 + 3*n - 1/36)), where A is the Glaisher-Kinkelin constant A074962.

A268504(n) * A294318(n) * A294319(n) = A000178(3*n + 2).

cp's OEIS Frontend

A098694 Double-superfactorials: a(n) = Product_{k=1..n} (2k)!.

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A268506 a(n) = Product_{k=0..n} (5*k)!.

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A268196 a(n) = Product_{k=0..n} binomial(3*k,k).

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A268505 a(n) = Product_{k=0..n} (4*k)!.

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A100732 a(n) = (3*n)!.

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A271946 a(n) = Product_{k=0..n} (6*k)!.

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A271947 a(n) = Product_{k=0..n} (7*k)!.

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