A368721 - OEIS

%I #8 Jan 04 2024 08:56:39

%S 1,3,940896,18425962131085183248,

%T 652934720004728520613911984092239003385856,

%U 433324200327440062759688153700055880769227264159137063987248492437306880000

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

%C 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).

%F 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...

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

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

%Y Cf. A324437, A368723.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jan 04 2024