A367542 -id:A367542 - cp's OEIS frontend

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3ij + j^2).

1, 5, 12100, 188898484500, 91554454518735288960000, 4263420404009649597344435073399120000000, 46073465749493255153019723901007197815549903333795840000000000

Offset: 0

Vaclav Kotesovec, Dec 10 2023

nonn

Cf. A367543, A324403, A367542, A079478^2, A368064, A368065, A368066.

Table[Product[i^2 + 3*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

a(n) ~ 5^(5*n*(n+1)/2 + 1/2) * n^(2*n^2 - 1) / (2 * Pi * exp(3*n^2) * phi^(sqrt(5)*(n*(n+1) + 1/6) - 1/2)), where phi = A001622 is the golden ratio.

A367670 a(n) = Product_{i=1..n, j=1..n} (i^8 + i^4*j^4 + j^8).

Original entry on oeis.org

3, 171714816, 9817265089769041882465383168, 351690857158733335833718073682368165890982417955022627663773696

Offset: 1

Vaclav Kotesovec, Nov 26 2023

nonn

Cf. A367542, A367543, A367550, A367668.

Table[Product[Product[i^8 + i^4*j^4 + j^8, {i, 1, n}], {j, 1, n}], {n, 1, 7}]

from math import prod, factorial
def A367670(n): return (prod((k:=j**4)**2+(m:=i**4)*(m+k) for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**4)**2*3**n # Chai Wah Wu, Nov 26 2023

a(n) = A367550(n) * A367668(n).
a(n) = A367542(n) * A367543(n) * A367668(n).
a(n) ~ c * 3^(3*n*(n+1)/2) * (2 + sqrt(3))^(sqrt(3)*n*(n+1)) * n^(8*n^2 - 2) / exp(12*n^2 - Pi*(1 + sqrt(3))*n*(n+1)/2), where c = 0.05091893538977858024246640150391280389386566805866250210433631511020673755...

cp's OEIS Frontend

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3ij + j^2).

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Formula

A367670 a(n) = Product_{i=1..n, j=1..n} (i^8 + i^4*j^4 + j^8).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Formula

cp's OEIS Frontend

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*i*j + j^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A367670 a(n) = Product_{i=1..n, j=1..n} (i^8 + i^4*j^4 + j^8).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Formula

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3ij + j^2).