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 1-4 of 4 results.

A294324 a(n) = Product_{k=0..n} (5*k + 2)!.

Original entry on oeis.org

2, 10080, 4828336128000, 1717378459351319052288000000, 1930334638180469242638816526565470371840000000000, 21019161870767674789722561439867977128887689291877548419973120000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 28 2017

Keywords

Crossrefs

Cf. A268506, A294319, A294321, A294323, A294325, A294326.

Programs

  • Mathematica
    Table[Product[(5*k + 2)!, {k, 0, n}] , {n, 0, 10}]
FoldList[Times,(5Range[0,10]+2)!] (* Harvey P. Dale, Aug 11 2024 *)

Formula

a(n) ~ 2^(n/2 + 6/5) * 5^(5*n^2/2 + 5*n + 7/3) * n^(5*n^2/2 + 5*n + 137/60) * Pi^(n/2 + 11/10) / (A^(1/5) * (1 + sqrt(5))^(1/10) * Gamma(1/5)^(2/5) * Gamma(2/5)^(4/5) * exp(15*n^2/4 + 5*n - 1/60)), where A is the Glaisher-Kinkelin constant A074962.
A268506(n) * A294323(n) * A294324(n) * A294325(n) * A294326(n) = A000178(5*n+4).

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

Views

Author

Vaclav Kotesovec, Oct 28 2017

Keywords

Crossrefs

Cf. A168467, A268504, A294319.

Programs

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

Formula

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).

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

Original entry on oeis.org

2, 1440, 5225472000, 455547719673446400000, 2916586742141623158009180979200000000, 3278245620793706216637861108629164518335840256000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 28 2017

Keywords

Crossrefs

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

Programs

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

Formula

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).

A365266 a(n) = Product_{k=1..n} Gamma(6*k).

Original entry on oeis.org

1, 120, 4790016000, 1703748471578689536000000, 44045334006101976766560297729172439040000000000, 389438360216723307909581902233109465138002465491175688781168640000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 01 2023

Keywords

Crossrefs

Cf. A000178, A168467, A294319, A294322, A294326.
Cf. A055462, A306635, A306651.

Programs

  • Mathematica
    Table[Product[Gamma[6*k], {k, 1, n}], {n, 0, 10}]
Table[Product[(6*k-1)!, {k, 1, n}], {n, 0, 10}]

Formula

a(n) = A^(35/6) * exp(-35/72) * Gamma(1/3)^(5/3) * 2^(-125/72 + 3*n^2) * 3^(47/72 + 5*n/2 + 3*n^2) * Pi^(-25/12 - 5*n/2) * BarnesG(1 + n) * BarnesG(7/6 + n) * BarnesG(4/3 + n) * BarnesG(3/2 + n) * BarnesG(5/3 + n) * BarnesG(11/6 + n), where A = A074962 is the Glaisher-Kinkelin constant.
a(n) ~ A^(-1/6) * Gamma(1/3)^(5/3) * 2^(-35/72 + 3*n + 3*n^2) * 3^(47/72 + 5*n/2 + 3*n^2) * exp(1/72 - 5*n/2 - 9*n^2/2) * n^(19/72 + 5*n/2 + 3*n^2) * Pi^(-5/6 + n/2), where A = A074962 is the Glaisher-Kinkelin constant.
Showing 1-4 of 4 results.

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