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.

A272164 a(n) = Product_{k=0..n} (n^2-k)!.

Original entry on oeis.org

1, 1, 288, 53094139822080000, 7114507432973653690572666462301501337370624000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 21 2016

Keywords

Comments

The next term has 392 digits.

Crossrefs

Cf. A272095, A272163, A272238, A272241.

Programs

  • Mathematica
    Table[Product[(n^2-k)!, {k, 0, n}], {n, 0, 6}]
Table[BarnesG[n^2 + 2]/BarnesG[n^2 - n + 1], {n, 0, 6}]

Formula

a(n) = A272163(n) * ((n^2)!)^(n+1) / A272179(n)^n.
a(n) ~ exp(1/24 + n/6 - n^2 - n^3) * n^(1 + n^2 + 2*n^3) * (2*Pi)^((n+1)/2).

A272095 a(n) = Product_{k=0..n} binomial(n^2,k).

Original entry on oeis.org

1, 1, 24, 27216, 1956864000, 11593630125000000, 7004354761049263478784000, 515246658615545697034849051407876096, 5368556637668593177532650186945239827409750982656, 9038577429104951379916309583338181472480254559457860096000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2016

Keywords

Crossrefs

Cf. A272093, A272094, A272241.
Cf. A000178, A098694, A268196, A262261.

Programs

  • Mathematica
    Table[Product[Binomial[n^2, k], {k, 0, n}], {n, 0, 10}]
Table[((n^2)!)^(n+1) * BarnesG[n^2 - n + 1] / (BarnesG[n^2 + 2] * BarnesG[n+2]), {n, 0, 10}]

Formula

a(n) = ((n^2)!)^(n+1) / (A272164(n) * A000178(n)).
a(n) ~ A * exp(3*n^2/4 + 5*n/6 - 1/8) * n^(n^2/2 - 5/12) / (2*Pi)^((n+1)/2), where A = A074962 is the Glaisher-Kinkelin constant.

A272238 a(n) = Product_{k=0..n} (n^2+k)!.

Original entry on oeis.org

1, 2, 2073600, 25177856146146034974720000000, 14100949826093501607549529280892932893801777581548587107609477120000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2016

Keywords

Comments

The next term has 173 digits.

Crossrefs

Cf. A135860, A272164, A272180, A272237, A272241.

Programs

  • Mathematica
    Table[Product[(n^2+k)!, {k, 0, n}], {n, 0, 6}]
Table[BarnesG[n^2 + n + 2]/BarnesG[n^2 + 1], {n, 0, 6}]

Formula

a(n) = ((n^2+n)!)^(n+1) / A272237(n).
a(n) ~ exp(11/24 + n/6 - n^2 - n^3) * n^((1+n)*(1 + n + 2*n^2)) * (2*Pi)^((n+1)/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

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.
Showing 1-4 of 4 results.

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

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