A271947 -id:A271947 - 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

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.

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

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.

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.

A195391 a(n) = (7*n)!.

Original entry on oeis.org

1, 5040, 87178291200, 51090942171709440000, 304888344611713860501504000000, 10333147966386144929666651337523200000000, 1405006117752879898543142606244511569936384000000000, 608281864034267560872252163321295376887552831379210240000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A271947.

Magma
```
[Factorial(7*n): n in [0..10]];
```

(7*Range[0, 8])! (* Paolo Xausa, Aug 12 2025 *)

a(n)=(7*n)! \\ Charles R Greathouse IV, Dec 27 2011

A367571 a(n) = Product_{k=0..n} (7*k)! / k!^7.

Original entry on oeis.org

1, 5040, 3432645216000, 626489905645044080640000000, 41646279370357699257014919153469440000000000000, 1200992054275801322636044235924808416678612164215512865177600000000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

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

Cf. A000178, A071549, A271947.

Cf. A007685, A367567, A367568, A367569, A367570.

Table[Product[(7*k)!/k!^7, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[7*k,k] * Binomial[6*k,k] * Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(7*k,k) * binomial(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A271947(n) / A000178(n)^7.
a(n) ~ A^(48/7) * 7^(7*n^2/2 + 4*n - 1/84) * exp(3*n - 4/7) / (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) * n^(3*n + 29/14) * (2*Pi)^(3*n + 3/2)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

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

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

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

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

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A195391 a(n) = (7*n)!.

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A367571 a(n) = Product_{k=0..n} (7*k)! / k!^7.

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