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.

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

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).

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

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.

A370483 a(n) = Product_{k=0..n} binomial(n^2 + k^2, k^2).

Original entry on oeis.org

1, 2, 350, 347633000, 101143578356902991250, 422044560230008480282938965899488406272, 1208807563912714402070105775158111317516306396248661153276031151000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2).
a(n) = A371643(n) / ((n^2)!^(n+1) * A255322(n)).
a(n) ~ 2^(4*n^3/3 + n^2 + n/6 + 1/4) * exp((Pi-4)*n^3/3 + Pi*n/4) / (A255504 * n^(n + 1/2) * Pi^(n/2)).

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

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.