A324437 -id:A324437 - cp's OEIS frontend

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

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...

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...

A307215 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^4 + j^4)).

Original entry on oeis.org

1, 9, 4, 0, 7, 3, 0, 2, 8, 5, 3, 7, 2, 3, 6, 1, 5, 2, 9, 9, 5, 3, 8, 6, 0, 7, 7, 5, 9, 9, 6, 4, 7, 7, 7, 2, 0, 3, 8, 7, 0, 7, 9, 6, 8, 2, 9, 3, 2, 1, 7, 0, 9, 2, 8, 1, 3, 0, 6, 1, 3, 9, 7, 4, 7, 2, 5, 2, 2, 6, 4, 2, 1, 7, 2, 0, 7, 2, 8, 3, 4, 7, 5, 5, 8, 9, 5, 3, 1, 0, 6, 8, 7, 6, 7, 7, 0, 7, 0, 0, 5, 9, 6, 1, 4

Offset: 1

Vaclav Kotesovec, Mar 29 2019

Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).
Product_{i=1..n, j=1..n} (1 + 1/(i^2 + j^2)) = A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313315993144237973600...
Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = A307209 = 3.50478299933972837589112...

			1.94073028537236152995386077599647772038707968293217092813061397472522642172...

Cf. A307209, A324437.

(* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^4 + j^4), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]

Equals limit_{n->infinity} (Product_{i=1..n, j=1..n} (1 + i^4 + j^4)) / A324437(n).

A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3j + i^2j^2 - i*j^3 + j^4).

Original entry on oeis.org

1, 1936, 1765124816400, 19271059559619728900751360000, 25048411180596698786915756280274804766474649600000000, 23045227505577134384745253646275782332295626096040088365089618773238077194240000000000

Offset: 1

Vaclav Kotesovec, Nov 26 2023

nonn

Cf. A079478, A324437, A324438, A367550, A367668.

Table[Product[i^4 - i^3*j + i^2*j^2 - i*j^3 + j^4, {i, 1, n}, {j, 1, n}], {n, 1, 6}]

from math import prod, factorial
def A367679(n): return (prod(i*(i*(i*(i-j)+j**2)-j**3)+j**4 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2 # Chai Wah Wu, Nov 26 2023

a(n) = A324438(n) / A079478(n).

a(n) ~ c * n^(4*n^2 - 5/6) * phi^(sqrt(5)*n*(n+1)) / exp(6*n^2 - sqrt(phi)*Pi*n*(n+1)/5^(1/4)), where phi = A001622 is the golden ratio and c = 0.2505211390193028244009922677012518708897316924498037078191143761182342931773594...

cp's OEIS Frontend

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

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

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A307215 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^4 + j^4)).

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Mathematica

Formula

A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3j + i^2j^2 - i*j^3 + j^4).

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cp's OEIS Frontend

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

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A307215 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^4 + j^4)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3*j + i^2*j^2 - i*j^3 + j^4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Formula

A367679 a(n) = Product_{i=1..n, j=1..n} (i^4 - i^3j + i^2j^2 - i*j^3 + j^4).