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).

A294325 a(n) = Product_{k=0..n} (5*k + 3)!.

Original entry on oeis.org

6, 241920, 1506440871936000, 9644797427717007797649408000000, 249337464544494851133170653103408676989829120000000000, 76020086814652932482688746849816272353956621412690696880710462996480000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 28 2017

Keywords

Crossrefs

Cf. A268506, A294322, A294323, A294324, A294326.

Programs

  • Mathematica
    Table[Product[(5*k + 3)!, {k, 0, n}] , {n, 0, 10}]

Formula

a(n) ~ (1 + sqrt(5))^(2/5) * Gamma(2/5)^(1/5) * 2^(n/2 + 1/5) * 5^(5*n^2/2 + 6*n + 43/12) * n^(5*n^2/2 + 6*n + 203/60) * Pi^(n/2 + 3/5) / (A^(1/5) * Gamma(1/5)^(2/5) * exp(15*n^2/4 + 6*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).

A294323 a(n) = Product_{k=0..n} (5*k + 1)!.

Original entry on oeis.org

1, 720, 28740096000, 601322989968949248000000, 30722158107023001697205508762501120000000000, 12389984031943899068723274670059592852478855603111854080000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 28 2017

Keywords

Crossrefs

Cf. A168467, A268506, A294318, A294320, A294324, A294325, A294326.

Programs

  • Mathematica
    Table[Product[(5*k + 1)!, {k, 0, n}] , {n, 0, 10}]

Formula

a(n) ~ 2^(n/2 + 7/10) * 5^(5*n^2/2 + 4*n + 4/3) * n^(5*n^2/2 + 4*n + 83/60) * Pi^(n/2 + 3/5) * Gamma(2/5)^(1/5) / (A^(1/5) * (1 + sqrt(5))^(1/10) * Gamma(1/5)^(2/5) * exp(15*n^2/4 + 4*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).

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.

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

Content is available under The OEIS End-User License Agreement.