%I #9 Jan 06 2024 14:07:03

%S 1,3,30180180096,130911253854794147456410254996552949923277899497472

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

%C Next term is too long to be included.

%C 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).

%F 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...

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

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

%Y Cf. A324437, A368721.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jan 04 2024