A268505 -id:A268505 - 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

A100733 a(n) = (4*n)!.

Original entry on oeis.org

1, 24, 40320, 479001600, 20922789888000, 2432902008176640000, 620448401733239439360000, 304888344611713860501504000000, 263130836933693530167218012160000000, 371993326789901217467999448150835200000000, 815915283247897734345611269596115894272000000000

Offset: 0

Ralf Stephan, Dec 08 2004

From Karol A. Penson, Jun 11 2009: (Start)
Integral representation of a(n) as n-th moment of a positive function W(x) = (1/4)*exp(-x^(1/4))/x^(3/4) on the positive axis:
a(n) = Integral_{x=0..oo} x^n*W(x) dx = Integral_{x=0..oo} x^n*(1/4)*exp(-x^(1/4))/x^(3/4) dx, n >= 0.
This is the solution of the Stieltjes moment problem with the moments a(n), n >= 0.
As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.
This is indeed the case as by construction the following "doubly" infinite family:
V(k,a,x) = (1/4)*exp(-x^(1/4))*(a*sin((3/4)*Pi*k + tan((1/4)*Pi*k)*x^(1/4)) + 1)/x^(3/4),
with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).
(End)

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

Cf. A000142, A010050, A100732, A100734, A268505, A332890.

[Factorial(4*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

(4*Range[0,20])! (* Harvey P. Dale, Oct 03 2014 *)

E.g.f.: 1/(1-x^4).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).
Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)
Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - Amiram Eldar, Feb 14 2021

More terms from Harvey P. Dale, Oct 03 2014

A268504 a(n) = Product_{k=0..n} (3*k)!.

Original entry on oeis.org

1, 6, 4320, 1567641600, 750902834626560000, 981936389699695364014080000000, 6286723722110812136775527266768650240000000000, 321194638135877430211257700556824829511701622266265600000000000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

nonn

Partial products of A100732. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..22

Cf. A000178, A098694, A100732, A268196, A268505, A268506, A271946, A271947.

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

{a(n) = prod(k=1, n, (3*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ Gamma(1/3)^(1/3) * 3^(3*n^2/2 + 2*n + 11/36) * 2^(n/2 + 1/3) * Pi^(n/2 + 1/3) * n^(3*n^2/2 + 2*n + 19/36) / (A^(1/3) * exp(9*n^2/4 + 2*n - 1/36)), where A = A074962 is the Glaisher-Kinkelin constant.

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.

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.

A294322 a(n) = Product_{k=0..n} (4*k + 3)!.

Original entry on oeis.org

6, 30240, 1207084032000, 1578472848668491776000000, 192013488168893760607534429765632000000000, 4963935910233933921764132479991824059486720994836480000000000000

Offset: 0

Vaclav Kotesovec, Oct 28 2017

nonn

Cf. A268505, A294320, A294321, A294325.

Table[Product[(4*k + 3)!, {k, 0, n}] , {n, 0, 10}]

a(n) ~ 2^(4*n^2 + 23*n/2 + 47/6) * n^(2*n^2 + 11*n/2 + 173/48) * Pi^(n/2 + 1/4) * Gamma(1/4)^(1/2) / (A^(1/4) * exp(3*n^2 + 11*n/2 - 1/48)), where A is the Glaisher-Kinkelin constant A074962.

A268505(n) * A294320(n) * A294321(n) * A294322(n) = A000178(4*n + 3).

A294320 a(n) = Product_{k=0..n} (4*k + 1)!.

Original entry on oeis.org

1, 120, 43545600, 271159356948480000, 96447974277170077976494080000000, 4927617876373416030299815278723491640115200000000000, 76433315893700635598991132508610825923227961061372903345356800000000000000000

Offset: 0

Vaclav Kotesovec, Oct 28 2017

nonn

Cf. A168467, A268505, A294318, A294321, A294322.

Table[Product[(4*k + 1)!, {k, 0, n}] , {n, 0, 10}]

a(n) ~ 2^(4*n^2 + 15*n/2 + 10/3) * n^(2*n^2 + 7*n/2 + 65/48) * Pi^(n/2 + 3/4) / (A^(1/4) * Gamma(1/4)^(1/2) * exp(3*n^2 + 7*n/2 - 1/48)), where A is the Glaisher-Kinkelin constant A074962.

A268505(n) * A294320(n) * A294321(n) * A294322(n) = A000178(4*n + 3).

A294321 a(n) = Product_{k=0..n} (4*k + 2)!.

Original entry on oeis.org

2, 1440, 5225472000, 455547719673446400000, 2916586742141623158009180979200000000, 3278245620793706216637861108629164518335840256000000000000

Offset: 0

Vaclav Kotesovec, Oct 28 2017

nonn

Cf. A268505, A294319, A294320, A294322, A294324.

Table[Product[(4*k + 2)!, {k, 0, n}] , {n, 0, 10}]

a(n) ~ 2^(4*n^2 + 19*n/2 + 35/6) * n^(2*n^2 + 9*n/2 + 113/48) * Pi^(n/2 + 3/4) / (A^(1/4) * Gamma(1/4)^(1/2) * exp(3*n^2 + 9*n/2 - 1/48)), where A is the Glaisher-Kinkelin constant A074962.

A268505(n) * A294320(n) * A294321(n) * A294322(n) = A000178(4*n + 3).

cp's OEIS Frontend

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

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A100733 a(n) = (4*n)!.

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A268504 a(n) = Product_{k=0..n} (3*k)!.

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

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

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A294322 a(n) = Product_{k=0..n} (4*k + 3)!.