cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A296591 a(n) = Product_{k=0..n} (n + k)!.

Original entry on oeis.org

1, 2, 288, 12441600, 421382062080000, 23120161750363668480000000, 3683853104727992382799761899520000000000, 2777528195026874073410445622205453260145295360000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Links

Crossrefs

Cf. A001142, A007685, A086205, A098694, A110131, A112332, A268196, A296589, A296590.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) *(2*n-1)! *(2*n)! /(n-1)!)
    end:
seq(a(n), n=0..7);  # Alois P. Heinz, Jul 11 2024
  • Mathematica
    Table[Product[(n + k)!, {k, 0, n}], {n, 0, 10}]
Table[Product[(2*n - k)!, {k, 0, n}], {n, 0, 10}]
Table[BarnesG[2*n + 2]/BarnesG[n + 1], {n, 0, 10}]

Formula

a(n) = BarnesG(2*n + 2) / BarnesG(n + 1).
a(n) ~ 2^(2*n^2 + 5*n/2 + 11/12) * n^((n+1)*(3*n+1)/2) * Pi^((n+1)/2) / exp(9*n^2/4 + 2*n).

Extensions

Missing a(0)=1 inserted by Georg Fischer, Nov 18 2021

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

Views

Author

Vaclav Kotesovec, Apr 17 2016

Keywords

Comments

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.

Crossrefs

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

Programs

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

Formula

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.

A272094 a(n) = Product_{k=0..n} binomial(k^2,k).

Original entry on oeis.org

1, 1, 6, 504, 917280, 48735086400, 94925811409228800, 8154182636726616909619200, 36091760791026276649159689107865600, 9415901310649088228943246038670339934863360000, 162992165498634702043940163611264755298214594247272038400000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

Crossrefs

Cf. A255322, A272093, A272095.
Cf. A000178, A098694, A268196, A262261.

Programs

  • Mathematica
    Table[Product[Binomial[k^2, k], {k, 0, n}], {n, 0, 10}]

Formula

a(n) = A255322(n) / (A272168(n) * A000178(n)).
a(n) ~ c1/c2 * A * exp(-1/12 + n/2 + n^2/4) * n^(1/12 + n^2/2) / (2*Pi)^(n/2), where c1 = Product_{k>=1} (k^2)!/stirling(k^2) = 1.14426047263759216966268786..., c2 = Product_{k>=2} (k*(k-1))!/stirling(k*(k-1)) = 1.086533635964823338078329..., stirling(n) = sqrt(2*Pi*n) * n^n / exp(n) is the Stirling approximation of n!, and A = A074962 is the Glaisher-Kinkelin constant.

A272095 a(n) = Product_{k=0..n} binomial(n^2,k).

Original entry on oeis.org

1, 1, 24, 27216, 1956864000, 11593630125000000, 7004354761049263478784000, 515246658615545697034849051407876096, 5368556637668593177532650186945239827409750982656, 9038577429104951379916309583338181472480254559457860096000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

Crossrefs

Cf. A272093, A272094, A272241.
Cf. A000178, A098694, A268196, A262261.

Programs

  • Mathematica
    Table[Product[Binomial[n^2, k], {k, 0, n}], {n, 0, 10}]
Table[((n^2)!)^(n+1) * BarnesG[n^2 - n + 1] / (BarnesG[n^2 + 2] * BarnesG[n+2]), {n, 0, 10}]

Formula

a(n) = ((n^2)!)^(n+1) / (A272164(n) * A000178(n)).
a(n) ~ A * exp(3*n^2/4 + 5*n/6 - 1/8) * n^(n^2/2 - 5/12) / (2*Pi)^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant.

A272093 a(n) = Product_{k=0..n} binomial(k*n,k).

Original entry on oeis.org

1, 1, 12, 3780, 44844800, 26352845268750, 953083353075475894272, 2537540586421634737033298208000, 579150777545101402084349505293757972480000, 12933741941622730846344367593442776825612980847409218750, 31768605393074559234133528464091374346848946682424165820313600000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

Crossrefs

Cf. A272094, A272094, A272095.
Cf. A000178, A098694, A268196, A262261.

Programs

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

Formula

a(n) = A272096(n) / (A272166(n) * A000178(n)).
a(n) ~ A^2 * exp(n^2/2 + 3*n/4 + 1/12) * n^(n^2/2 - 1/3) / (2*Pi)^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant.

A296590 a(n) = Product_{k=0..n} binomial(2*n - k, k).

Original entry on oeis.org

1, 1, 3, 30, 1050, 132300, 61122600, 104886381600, 674943865596000, 16407885372638760000, 1515727634953623371280000, 534621388490302221024396480000, 722849817707190846398223943885440000, 3759035907022704558524683975387453632000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Comments

Apart from the offset the same as A203469. - R. J. Mathar, Alois P. Heinz, Jan 02 2018

Links

Crossrefs

Cf. A001142, A007685, A086205, A098694, A110131, A112332, A203471, A268196, A296589, A296591, A338550.

Programs

  • Maple
    A296590 := proc(n)
    mul( binomial(2*n-k,k),k=0..n) ;
end proc:
seq(A296590(n),n=0..7) ; # R. J. Mathar, Jan 03 2018
  • Mathematica
    Table[Product[Binomial[2*n-k, k], {k, 0, n}], {n, 0, 15}]
Table[Glaisher^(3/2) * 2^(n^2 - 1/24) * BarnesG[n + 3/2] / (E^(1/8) * Pi^(n/2 + 1/4) * BarnesG[n + 2]), {n, 0, 15}]

Formula

a(n) = A^(3/2) * 2^(n^2 - 1/24) * BarnesG(n + 3/2) / (exp(1/8) * Pi^(n/2 + 1/4) * BarnesG(n + 2)), where A is the Glaisher-Kinkelin constant A074962.
a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^((n+1)/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962.
Product_{1 <= j <= i <= n} (i + j - 1)/(i - j + 1). - Peter Bala, Oct 25 2024

A296607 a(n) = BarnesG(2*n).

Original entry on oeis.org

0, 1, 2, 288, 24883200, 5056584744960000, 6658606584104736522240000000, 127313963299399416749559771247411200000000000, 69113789582492712943486800506462734562847413501952000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Links

Crossrefs

Cf. A000178, A098694, A296591, A296608, A306635.

Programs

  • Mathematica
    Table[BarnesG[2*n], {n, 0, 10}]
Table[Glaisher^3 * E^(-1/4) * 2^(2*n^2 - 3*n + 11/12) * Pi^(1/2 - n) * BarnesG[n] * BarnesG[n + 1/2]^2 * BarnesG[n+1], {n, 0, 10}]

Formula

a(n) = A^3 * exp(-1/4) * 2^(2*n^2 - 3*n + 11/12) * Pi^(1/2 - n) * BarnesG(n) * BarnesG(n + 1/2)^2 * BarnesG(n+1), where A is the Glaisher-Kinkelin constant A074962.
a(n) ~ 2^(2*n^2 - n - 1/12) * exp(1/12 + 2*n - 3*n^2) * n^(2*n^2 - 2*n + 5/12) * Pi^(n - 1/2) / A, where A is the Glaisher-Kinkelin constant A074962.
a(n) = A000178(2*n-2), n>0. - R. J. Mathar, Jul 24 2025

A272096 a(n) = Product_{k=0..n} (k*n)!.

Original entry on oeis.org

1, 1, 48, 1567641600, 9698137182219213471744000000, 21488900044302744250061179567064173417691432878080000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

Comments

The next term has 126 digits.

Crossrefs

Cf. A000178, A098694, A268504, A268505, A268506, A271946.
Cf. A255322, A272093.

Programs

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

Formula

a(n) ~ A^n * n^(1/4 + 13*n/12 + n^2 + n^3) * (2*Pi)^(1/4 + n/2) / exp(n*(2 + 2*n + 3*n^2)/4), where A = A074962 is the Glaisher-Kinkelin constant.
Previous Showing 11-18 of 18 results.

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