A307215 - OEIS

%I #11 Mar 29 2019 06:43:37

%S 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,

%T 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,

%U 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

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

%C Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

%C 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...

%C Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)) = A307209 = 3.50478299933972837589112...

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

%e 1.94073028537236152995386077599647772038707968293217092813061397472522642172...

%t (* 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}]

%Y Cf. A307209, A324437.

%K nonn,cons

%O 1,2

%A _Vaclav Kotesovec_, Mar 29 2019