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 1-5 of 5 results.

A324403 a(n) = Product_{i=1..n, j=1..n} (i^2 + j^2).

Original entry on oeis.org

1, 2, 400, 121680000, 281324160000000000, 15539794609114833408000000000000, 49933566483104048708063697937367040000000000000000, 19323883089768863178599626514889213871887405416448000000000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 26 2019

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A079478, A324426, A324437, A324438, A324439, A324440, A367834.
Cf. A272244, A293290, A324402, A324443, A324425.

Programs

  • Maple
    a:= n-> mul(mul(i^2+j^2, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i^2+j^2, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
  • PARI
    a(n) = prod(i=1, n, prod(j=1, n, i^2+j^2)); \\ Michel Marcus, Feb 27 2019
  • Python
    from math import prod, factorial
def A324403(n): return (prod(i**2+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2<Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ 2^(n*(n+1) - 3/4) * exp(Pi*n*(n+1)/2 - 3*n^2 + Pi/12) * n^(2*n^2 - 1/2) / (Pi^(1/4) * Gamma(3/4)).
a(n) = 2*n^2*a(n-1)*Product_{i=1..n-1} (n^2 + i^2)^2. - Chai Wah Wu, Feb 26 2019
For n>0, a(n)/a(n-1) = A272244(n)^2 / (2*n^6). - Vaclav Kotesovec, Dec 02 2023
a(n) = exp(2*Integral_{x=0..oo} (n^2/(x*exp(x)) - (cosh(n*x) - cos(n*x))/(x*exp((n + 1)*x)*(cosh(x) - cos(x)))) dx)/2^(n^2). - Velin Yanev, Jun 30 2025

Extensions

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

A367958 a(n) = Product_{i=1..n, j=1..n} (i + 5*j).

Original entry on oeis.org

1, 6, 5544, 2822916096, 1723467782592331776, 2210440498434925488635904000000, 9234659938893939743399592700454853672960000000, 180150216814109052335771891722360520401032374209013927116800000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 06 2023

Keywords

Comments

In general, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^(1/d) * (Product_{j=1..d} Gamma(j/d)^(j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*n*(n+1) + (d+1)^2/(12*d) + 1/12) * n^(n^2 - d/12 - 1/4 - 1/(12*d)) / ((2*Pi)^((d+1)/4) * exp(3*n^2/2 + 1/(12*d)) * d^((n*(d*n + (d+1)))/2 - 1/(12*d))), where A = A074962 is the Glaisher-Kinkelin constant.
Equivalently, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^d * (Product_{j=1..d} BarnesG(j/d)) * (2*Pi)^((d-3)/4) * (d+1)^((d + (d+1)^2*(6*n*(n+1) + 1)) / (12*d)) * n^(n^2 - 1/4 - 1/(12*d) - d/12) / (d^((n+1)*(d*n + 1)/2) * exp(3*n^2/2 + d/12)).

Crossrefs

Cf. A079478 (d=1), A324402 (d=2), A367956 (d=3), A367957 (d=4).
Cf. A074962, A367842, A367944.

Programs

  • Maple
    a:= n-> mul(mul(i+5*j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Dec 06 2023
  • Mathematica
    Table[Product[i + 5*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

Formula

a(n) ~ A^(1/5) * (1 + sqrt(5))^(1/10) * 2^(18*n*(n+1)/5 + 29/60) * 3^(18*n*(n+1)/5 + 41/60) * n^(n^2 - 41/60) / (Pi^(1/10) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(n*(5*n+6)/2 + 1/3) * exp(3*n^2/2 + 1/60)), where A = A074962 is the Glaisher-Kinkelin constant.

A367956 a(n) = Product_{i=1..n, j=1..n} (i + 3*j).

Original entry on oeis.org

1, 4, 1120, 79833600, 3173289799680000, 123650071173117090201600000, 7337799401269093351612002462597120000000, 951792703318385182295191545713146608287219712000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 06 2023

Keywords

Crossrefs

Cf. A074962, A079478, A324402, A367957, A367958.

Programs

  • Mathematica
    Table[Product[i + 3*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

Formula

a(n) ~ A^(1/3) * 2^(16*n*(n+1)/3 + 13/18) * n^(n^2 - 19/36) / (Pi^(1/3) * Gamma(1/3)^(1/3) * 3^(n*(3*n+4)/2 + 11/36) * exp(3*n^2/2 + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant.

A367957 a(n) = Product_{i=1..n, j=1..n} (i + 4*j).

Original entry on oeis.org

1, 5, 2700, 567567000, 101370917007360000, 26995322179162164731904000000, 16635639072295355604762223305031680000000000, 34026881962001914598329145027742925521204742717440000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 06 2023

Keywords

Crossrefs

Cf. A074962, A079478, A324402, A367956, A367958.

Programs

  • Mathematica
    Table[Product[i + 4*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

Formula

a(n) ~ A^(1/4) * 5^(25*n*(n+1)/8 + 29/48) * n^(n^2 - 29/48) / (Pi^(1/4) * Gamma(1/4)^(1/2) * 2^(n*(4*n+5) + 5/6) * exp(3*n^2/2 + 1/48)), where A = A074962 is the Glaisher-Kinkelin constant.

A368068 a(n) = Product_{i=1..n, j=1..n} (2*i^2 + 5*i*j + 2*j^2).

Original entry on oeis.org

1, 9, 129600, 40327580160000, 1311346674278439321600000000, 13821139470331790817454891043295068160000000000, 114180111981355345833797461507302737916551512227408406118400000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A324402, A368065.

Programs

  • Mathematica
    Table[Product[2*i^2 + 5*i*j + 2*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) = Product_{i=1..n, j=1..n} (i + 2*j) * (2*i + j).
a(n) = A324402(n)^2.
a(n) ~ A * 3^(9*n*(n+1)/2 + 11/12) * n^(2*n^2 - 11/12) / (Pi * 2^(2*n^2 + 3*n + 17/12) * exp(3*n^2 + 1/12)), where A is the Glaisher-Kinkelin constant A074962.
Showing 1-5 of 5 results.

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