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%I A367944 #10 Dec 08 2023 04:41:44
%S A367944 1,6,27216,1344924798336,3605580335899213007486976,
%T A367944 1648055031941075082958467426002632704000000,
%U A367944 312704667066499295437237787452750428210311485710262201221120000000
%N A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).
%C A367944 In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).
%C A367944 c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.
%F A367944 a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...
%t A367944 Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]
%Y A367944 Cf. A324403 (d=1), A367941 (d=2), A367942 (d=3), A367943 (d=4).
%Y A367944 Cf. A367542, A367543, A367958.
%K A367944 nonn
%O A367944 0,2
%A A367944 _Vaclav Kotesovec_, Dec 05 2023

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