A368720 -id:A368720 - cp's OEIS frontend

A368622 a(n) = Product_{j=1..n, k=1..n} (j^2 + k^2 + n^2).

1, 3, 5832, 172907569296, 419358815743567702818816, 267800543010963952830647446563000000000000, 110831581527076064529150462985910455129725821244148698662830080000

Offset: 0

Vaclav Kotesovec, Jan 01 2024

nonn

The limit has a closed form. In Mathematica: Exp[Integrate[Log[x^2 + y^2 + 1], {x,0,1}, {y,0,1}]]. The output is extremely large.

Cf. A272244, A324403, A324425, A368720, A368721.

Table[Product[j^2 + k^2 + n^2, {j, 1, n}, {k, 1, n}], {n, 0, 10}]

Limit_{n->oo} a(n)^(1/(n^2)) / n^2 = exp(Integral_{x=0..1, y=0..1} log(x^2 + y^2 + 1) dy dx) = 1.6143980185761253961882683158432481977126507900460725431661...

A368722 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^3 + j^3 + k^3).

Original entry on oeis.org

1, 3, 353736000, 4795587079853800623303366670123008000000

Offset: 0

Vaclav Kotesovec, Jan 04 2024

nonn

Next term is too long to be included.

Cf. A306594, A324425, A368723.

Cf. A324426, A368720.

Table[Product[i^3 + j^3 + k^3, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 0, 5}]

Limit_{n->oo} a(n)^(1/(n^3)) / n^3 = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^3 + y^3 + z^3) dz dy dx) = 0.5334736919092639993380174031245...

A368721 a(n) = Product_{j=1..n, k=1..n} (j^4 + k^4 + n^4).

Original entry on oeis.org

1, 3, 940896, 18425962131085183248, 652934720004728520613911984092239003385856, 433324200327440062759688153700055880769227264159137063987248492437306880000

Offset: 0

Vaclav Kotesovec, Jan 04 2024

nonn

In general, for m>0, limit_{n->oo} (Product_{j=1..n, k=1..n} (j^m + k^m + n^m))^(1/(n^2)) / n^m = exp(Integral_{x=0..1, y=0..1} log(x^m + y^m + 1) dy dx) = 3 / exp(HurwitzLerchPhi(-1/2, 1, 1 + 1/m)/2 + Integral_{x=0..1} HurwitzLerchPhi(-1/(1 + x^m), 1, 1 + 1/m) / (1 + x^m) dx).

Cf. A368685 (m=1), A368622 (m=2), A368720 (m=3).

Cf. A324437, A368723.

Table[Product[j^4 + k^4 + n^4, {j, 1, n}, {k, 1, n}], {n, 0, 6}]

Limit_{n->oo} a(n)^(1/(n^2)) / n^4 = exp(Integral_{x=0..1, y=0..1} log(x^4 + y^4 + 1) dy dx) = 1.35451345305131009729671041498902524074679186355643287514556358...

cp's OEIS Frontend

A368622 a(n) = Product_{j=1..n, k=1..n} (j^2 + k^2 + n^2).

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A368722 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^3 + j^3 + k^3).

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Author

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Programs

Mathematica

Formula

A368721 a(n) = Product_{j=1..n, k=1..n} (j^4 + k^4 + n^4).

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Author

Keywords

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Crossrefs

Programs

Mathematica

Formula