A368721 -id:A368721 - 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...

A368723 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^4 + j^4 + k^4).

Original entry on oeis.org

1, 3, 30180180096, 130911253854794147456410254996552949923277899497472

Offset: 0

Vaclav Kotesovec, Jan 04 2024

nonn

Next term is too long to be included.

In general, for m>0, limit_{n->oo} (Product_{i=1..n, j=1..n, k=1..n} (i^m + j^m + k^m))^(1/(n^3)) / n^m = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^m + y^m + z^m) dz dy dx) = exp(Integral_{x=0..1, y=0..1} (log(1 + x^k + y^k) - k + k*hypergeom2F1(1/k, 1, (k+1)/k, -1/(x^k + y^k))) dy dx).

Cf. A306594 (m=1), A324425 (m=2), A368722 (m=3).

Cf. A324437, A368721.

Table[Product[i^4 + j^4 + k^4, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 0, 5}]

Limit_{n->oo} a(n)^(1/(n^3)) / n^4 = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^4 + y^4 + z^4) dz dy dx) = 0.3570458697635761757481417...

A368720 a(n) = Product_{j=1..n, k=1..n} (j^3 + k^3 + n^3).

Original entry on oeis.org

1, 3, 69360, 1522177267723200, 391047538356893112890665992192000, 6785985430272886334677590546861463643140253138288640000000

Offset: 0

Vaclav Kotesovec, Jan 04 2024

nonn

Cf. A368685, A368622, A368721.

Cf. A324426, A368722.

Table[Product[j^3 + k^3 + n^3, {j, 1, n}, {k, 1, n}], {n, 0, 6}]

Limit_{n->oo} a(n)^(1/(n^2)) / n^3 = exp(Integral_{x=0..1, y=0..1} log(x^3 + y^3 + 1) dy dx) = 1.4501318426779115107161463152054835452511102493859148975306452...

cp's OEIS Frontend

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

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A368723 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^4 + j^4 + k^4).

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Author

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Programs

Mathematica

Formula

A368720 a(n) = Product_{j=1..n, k=1..n} (j^3 + k^3 + n^3).

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Mathematica

Formula