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 1-3 of 3 results.

A306760 a(n) = Product_{i=1..n, j=1..n} (i*j + 1).

Original entry on oeis.org

1, 2, 90, 705600, 4105057320000, 52487876090562232320000, 3487017405172854771910634342400000000, 2448893405298238642974553493547144534294528000000000000, 33257039167768610289435138215602132823918399655132218973388800000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 08 2019

Keywords

Crossrefs

Programs

  • Maple
    a:= n-> mul(mul(i*j+1, i=1..n), j=1..n):
    seq(a(n), n=0..9);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i*j + 1, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
    Table[n!^(2*n) * Product[Binomial[n + 1/j, n], {j, 1, n}], {n, 1, 10}]

Formula

a(n) ~ c * n^(n*(2*n+1) + 2*gamma) * (2*Pi)^n * exp(1/6 + log(n)^2 - 2*n^2), where c = 1/A306765 and gamma is the Euler-Mascheroni constant A001620.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324590 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

Original entry on oeis.org

1, 2, 1080, 16133644800, 139256878046022696960000, 6288402750181849898716908922601472000000000, 8322157105451357856813375261666887975745751468393837363200000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

Crossrefs

Programs

  • Maple
    a:= n-> n!^(4*n)*mul(binomial(n+1/k^2, n), k=1..n):
    seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[n!^(4*n) * Product[Binomial[1/k^2 + n, n], {k, 1, n}], {n, 1, 8}]

Formula

a(n) ~ n!^(4*n) * n^(Pi^2/6) / A303670.
a(n) ~ n^(4*n^2 + 2*n + Pi^2/6) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.
a(n) = n!^n * A324596(n).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324596 a(n) = n!^(3*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

Original entry on oeis.org

1, 2, 270, 74692800, 419731620267960000, 252716802910471719823692648960000, 59736659298524125157504488525739821430187940800000000, 16079377413231597423103950774423398920424350187193326745026311068057600000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

Crossrefs

Programs

  • Maple
    a:= n-> n!^(3*n)*mul(binomial(n+1/k^2, n), k=1..n):
    seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[n!^(3*n) * Product[Binomial[n + 1/k^2, n], {k, 1, n}], {n, 1, 8}]

Formula

a(n) ~ n!^(3*n) * n^(Pi^2/6) / A303670.
a(n) ~ n^(3*n*(2*n+1)/2 + Pi^2/6) * (2*Pi)^(3*n/2) / exp(3*n^2 - 1/4 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023
Showing 1-3 of 3 results.