A368622 -id:A368622 - cp's OEIS frontend

A324425 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^2 + j^2 + k^2).

1, 3, 5668704, 550388591715704109656479285248, 152455602303300418998634460043817052571893573096619261814850281699755319515987050496

Offset: 0

Vaclav Kotesovec, Feb 27 2019

nonn

(a(n)^(1/n^3))/n^2 tends to 0.828859579669279... = A306617.

Cf. A079478, A306594, A368722, A368723.

Cf. A324403, A306617, A368622, A368623.

a:= n-> mul(mul(mul(i^2+j^2+k^2, i=1..n), j=1..n), k=1..n):
seq(a(n), n=0..5);  # Alois P. Heinz, Jun 24 2023

Table[Product[i^2+j^2+k^2, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 1, 6}]
Clear[a]; a[n_] := a[n] = If[n == 1, 3, a[n-1] * Product[k^2 + j^2 + n^2, {j, 1, n}, {k, 1, n}]^3 * (3*n^2) / (Product[k^2 + 2*n^2, {k, 1, n}]^3)]; Table[a[n], {n, 1, 6}] (* Vaclav Kotesovec, Mar 27 2019 *)

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

A368685 a(n) = Product_{j=1..n, k=1..n} (j + k + n).

Original entry on oeis.org

1, 3, 600, 35562240, 1434015830016000, 70448433354492434841600000, 6610702315560389323908439364075520000000, 1709479709147705756603303596364188306401499545600000000000, 1660017838341811463102474357555838707949172571314554168163386261504000000000000

Offset: 0

Vaclav Kotesovec, Jan 03 2024

nonn

Cf. A079478, A306594, A324444, A368622, A368686.

Table[Product[i+j+n, {i, 1, n}, {j, 1, n}], {n, 0, 8}]
Join[{1}, Table[3*BarnesG[n] * BarnesG[3*n] * Gamma[n]^2 * Gamma[3*n]^2 / (4*BarnesG[2*n]^2 * Gamma[2*n]^4), {n, 1, 8}]]

For n>0, a(n) = 3*BarnesG(n) * BarnesG(3*n) * Gamma(n)^2 * Gamma(3*n)^2 / (4*BarnesG(2*n)^2 * Gamma(2*n)^4).

a(n) ~ 3^(9*n^2/2 + 3*n + 5/12) * n^(n^2) / (2^(4*n^2 + 4*n + 5/6) * exp(3*n^2/2)).

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...

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...

A368686 a(n) = Product_{j=0..n, k=0..n} (j + k + n).

Original entry on oeis.org

0, 12, 172800, 1536288768000, 16189465114548633600000, 322110526445545505917029580800000000, 17555281051920416386104936570114748195012608000000000, 3580285185706909590176164870311607533516764550107699116769280000000000000

Offset: 0

Vaclav Kotesovec, Jan 03 2024

nonn

Cf. A079478, A306594, A324444, A368622, A368685.

Table[Product[i+j+n, {i, 0, n}, {j, 0, n}], {n, 0, 8}]
Join[{0}, Table[3*n*BarnesG[n] * BarnesG[3*n] * Gamma[3*n]^2 / BarnesG[2*n+1]^2, {n, 1, 8}]]

For n>0, a(n) = 3*n*BarnesG(n) * BarnesG(3*n) * Gamma(3*n)^2 / BarnesG(2*n+1)^2.
a(n) ~ 3^(9*n^2/2 + 3*n + 5/12) * n^((n+1)^2) / (2^(4*n^2 - 1/6) * exp(3*n^2/2 + 2*n)).
a(n) = 4*n*Gamma(2*n)^2 * A368685(n) / Gamma(n)^2.

cp's OEIS Frontend

A324425 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^2 + j^2 + k^2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Extensions

A368685 a(n) = Product_{j=1..n, k=1..n} (j + k + n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A368686 a(n) = Product_{j=0..n, k=0..n} (j + k + n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula