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.

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.

A371642 a(n) = Product_{k=0..n} (n^2 + k^2)! / (n^2 - k^2)!.

Original entry on oeis.org

1, 2, 806400, 29900785676206001356800000, 1118776785681133797769642926006209350326602179759885516800000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Cf. A272241, A371624, A371643.

Programs

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

Formula

a(n) = A371643(n) / A371624(n).
a(n) ~ c * 2^(n^2 - n/6 + 1/4) * exp((3*Pi-10)*n^3/9 - n^2 + Pi*n/4) * n^(4*n^3/3 + 2*n^2 + n/2 + 3/4) / A^(2*n), where c = 1.941002... = A255504 * (c from A371603) and A is the Glaisher-Kinkelin constant A074962.

A371646 a(n) = Product_{k=0..n} binomial(n^3, k^3).

Original entry on oeis.org

1, 1, 8, 59942025, 239830737497318918172122578944, 788243862228623056807478850630904903414781894638966172447366478063616699218750
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Cf. A001142, A371603.
Cf. A007685, A255358, A371468.

Programs

  • Mathematica
    Table[Product[Binomial[n^3, k^3], {k, 0, n}], {n, 0, 6}]

Formula

a(n) ~ c * exp((9/4 - sqrt(3)*Pi/8)*n^4 + (3*zeta(3)/(4*Pi^2) - Pi/(4*sqrt(3)) + 3)*n) / ((2*Pi)^(n/2) * A^(3*n^2) * 3^(9*n^4/8 - n^2/4 + 3*n/4) * n^(n^2/4 + 3*n/2 - 8/15)), where c = 0.498332919... and A is the Glaisher-Kinkelin constant A074962.

A371644 a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2 - k^2).

Original entry on oeis.org

1, 1, 10, 57915, 8235313944000, 1077099640691257742845893750, 4629575796245443900868634734946423885068807034000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Cf. A371603, A371624, A371642, A371643, A371645.

Programs

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

Formula

a(n) = A371642(n) / A371645(n).
a(n) = A371643(n) / (A371624(n) * A371645(n)).
a(n) ~ c * exp(Pi*n^3/3 + Pi*n/4 + n) / (2^(2*n^3/3 + 3*n/2) * Pi^(n/2) * A^(2*n) * n^(7*n/6 - 1/4)), where c = 0.761512... = 2^(1/4) * A255504 * (c from A371603) / (c from A371645) and A is the Glaisher-Kinkelin constant A074962.
Showing 1-4 of 4 results.

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