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A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

%I A307209 #26 Mar 29 2019 06:44:08
%S A307209 3,5,0,4,7,8,2,9,9,9,3,3,9,7,2,8,3,7,5,8,9,1,1,2,0,5,7,0,4,3,8,0,6,1,
%T A307209 2,5,5,8,3,8,9,3,2,4,7,8,6,2,7,1,2,7,5,3,5,4,1,9,9,4,6,2,6,6,1,4,0,5,
%U A307209 8,3,8,5,0,3,5,0,3,4,7,5,6,3,5,2,7,4,7,5,0,9,5,0,5,1,3,7,8,9,1,7,8,4,5,9,7
%N A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).
%C A307209 Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.
%C A307209 A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...
%C A307209 Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).
%H A307209 Vaclav Kotesovec, <a href="/A307209/b307209.txt">Table of n, a(n) for n = 1..224</a>
%F A307209 Equals limit_{n->infinity} A307210(n) / A324426(n).
%e A307209 3.50478299933972837589112057043806125583893247862712753541994626614058385...
%t A307209 (* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^3 + j^3), {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}]
%o A307209 (PARI) default(realprecision, 50); exp(sumalt(k=1, -(-1)^k/k*sumnum(i=1, sumnum(j=1, 1/(i^3+j^3)^k)))) \\ 15 decimals correct
%Y A307209 Cf. A073017, A307210, A307215, A324403, A324426, A324443.
%K A307209 nonn,cons
%O A307209 1,1
%A A307209 _Vaclav Kotesovec_, Mar 28 2019