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.

Showing 1-9 of 9 results.

A296589 a(n) = Product_{k=0..n} binomial(2*n, k).

Original entry on oeis.org

1, 2, 24, 1800, 878080, 2857680000, 63117561830400, 9577928124440387712, 10077943267571584204800000, 74054886893191804566576837427200, 3822038592032831128918160803430400000000, 1391938996758770867922655936144556115037409280000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

Links

Crossrefs

Cf. A001142, A007685, A086205, A110131, A112332, A268196, A296590, A296591.

Programs

  • Mathematica
    Table[Product[Binomial[2*n, k], {k, 0, n}], {n, 0, 12}]
Table[((2*n)!)^(n+1) / (n! * BarnesG[2*n + 2]), {n, 0, 12}]

Formula

a(n) = ((2*n)!)^(n+1) / (n! * BarnesG(2*n + 2)).
a(n) ~ A * exp(n^2 + n - 1/24) / (2^(5/12) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.

Extensions

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

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

A371603 a(n) = Product_{k=0..n} binomial(n^2, k^2).

Original entry on oeis.org

1, 1, 4, 1134, 333132800, 1319947441510156250, 876533819183888230348458418944000, 1185269534290897564185384010731432113450477770983533184
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 30 2024

Keywords

Crossrefs

Cf. A001142, A255322, A272095, A296591, A362288, A371624, A371646.

Programs

  • Mathematica
    Table[Product[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 8}]

Formula

a(n) = (n^2)!^(n+1) / (A255322(n) * A371624(n)).
a(n) ~ c * exp(2*n*(2*n^2/3 + 1)) / (A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(7*n/6 - 1/4)), where c = 0.6367427... and A is the Glaisher-Kinkelin constant A074962.

A371624 a(n) = Product_{k=0..n} (n^2 - k^2)!.

Original entry on oeis.org

1, 1, 144, 1755758592000, 66052111513207347990207922176000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 30 2024

Keywords

Comments

The next term has 88 digits.

Crossrefs

Cf. A001142, A255322, A296591, A362288, A371603.

Programs

  • Mathematica
    Table[Product[((2*n-k)*k)!, {k, 0, n}], {n, 0, 6}]
Table[Product[(n^2 - k^2)!, {k, 0, n}], {n, 0, 6}]

Formula

a(n) = (n^2)!^(n+1) / (A255322(n) * A371603(n)).
a(n) ~ c * A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(4*n^3/3 + n^2 + 5*n/6 + 1/4) / exp(16*n^3/9 + n^2/2 + n), where c = 1.291409... = sqrt(2*Pi) / (A255504 * c from A371603) and A is the Glaisher-Kinkelin constant A074962.

A371643 a(n) = Product_{k=0..n} (n^2 + k^2)!.

Original entry on oeis.org

1, 2, 116121600, 52498561358549216844165257625600000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Comments

The next term has 107 digits.

Crossrefs

Cf. A296591, A371468, A371624.

Programs

  • Mathematica
    Table[Product[(n^2+k^2)!, {k, 0, n}], {n, 0, 5}]

Formula

a(n) ~ 2^(4*n^3/3 + n^2 + 7*n/6 + 3/4) * exp(-26*n^3/9 + Pi*n^3/3 - 3*n^2/2 + Pi*n/4 - n) * Pi^((n+1)/2) * n^(8*n^3/3 + 3*n^2 + 4*n/3 + 1).

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

A371468 a(n) = Product_{k=0..n} (n^3 + k^3)!.

Original entry on oeis.org

1, 2, 306128067620555980800000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 01 2024

Keywords

Comments

The next term a(3) has 169 digits.

Crossrefs

Cf. A296591, A371643, A371646.

Programs

  • Mathematica
    Table[Product[(n^3 + k^3)!, {k, 0, n}], {n, 0, 5}]

Formula

a(n) ~ 2^(2*n^4 + n^3 + n^2/4 + 3*n/2 + 89/120) * Pi^((n+1)/2) * exp(Pi*sqrt(3)*n^4/4 - 59*n^4/16 - 3*n^3/2 - 3*n/2 + Pi*n/(2*sqrt(3)) - 9/320) * n^(15*n^4/4 + 9*n^3/2 + 3*n^2/4 + 3*n/2 + 3/2).

A374574 a(n) = Sum_{j=n..2n} j!.

Original entry on oeis.org

1, 3, 32, 870, 46224, 4037880, 522956160, 93928267440, 22324392518400, 6780385526302080, 2561327494111411200, 1177652997443424902400, 647478071469567800985600, 419450149241406188889984000, 316196664211373618844934963200, 274410818470142134209609852672000
Offset: 0

Views

Author

Alois P. Heinz, Jul 11 2024

Keywords

Links

Crossrefs

Row sums of A143084.
Cf. A000142, A100822, A143122, A296591 (the same for product).
Diagonal of A054115, A211370.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, [1, 3, 32][n+1],
     ((16*n^3-16*n^2-n+2)*a(n-1)-(n-1)*(16*n^3-20*n^2+6*n-1)
      *a(n-2)+2*(2*n-1)*(4*n+1)*(n-1)*(n-2)*a(n-3))/(4*n-3))
    end:
seq(a(n), n=0..15);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) -(n-1)! +(2*n-1)! +(2*n)!)
    end:
seq(a(n), n=0..15);

Formula

a(n) = a(n-1) - (n-1)! + (2*n-1)! + (2*n)! with a(0) = 1.
a(n) = Sum_{j=0..n} (n + j)!.
a(n) = A100822(2n,n).
a(n) = A143122(2n,n).

A372116 a(n) = Product_{k=0..n} (n+k)!^k.

Original entry on oeis.org

1, 2, 3456, 128994508800000, 21048441369734473363614597120000000000, 13080442484467245346116306952031286205761554346416540536012800000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 19 2024

Keywords

Comments

The next term has 146 digits.

Crossrefs

Cf. A055462, A255269, A296591, A367492, A368132.

Programs

  • Mathematica
    Table[Product[(n + k)!^k, {k, 0, n}], {n, 0, 8}]

Formula

a(n) ~ 2^(2*n^3/3 + 5*n^2/4 + 2*n/3 + 1/24) * Pi^(n*(n+1)/4) * n^(5*n^3/6 + 5*n^2/4 + 5*n/12) / exp(31*n^3/36 + 7*n^2/8 - 1/24).
For n>=1, a(n) = a(n-1) * A368132(n) * (2*n-1)!^n.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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