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A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

%I A306765 #28 Jul 01 2023 15:19:35
%S A306765 2,0,3,4,4,4,8,9,4,5,4,8,7,6,1,6,4,7,7,9,8,0,3,5,5,5,3,1,8,8,6,9,0,2,
%T A306765 6,3,5,5,9,7,9,4,3,9,8,6,3,7,0,2,3,7,6,2,6,0,0,0,5,2,8,4,1,6,5,6,5,0,
%U A306765 0,7,8,2,7,7,5,7,1,1,3,2,4,4,5,0,2,6,5,0,4,0,6,1,3,5,0,7,5,0,2,9,1,2,7,1,4
%N A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).
%H A306765 Vaclav Kotesovec, <a href="/A306765/b306765.txt">Table of n, a(n) for n = 1..297</a>
%F A306765 Equals exp(-gamma^2 + Sum_{j>=2} (-1)^j*Zeta(j)^2/j), where gamma is the Euler-Mascheroni constant A001620.
%F A306765 Equals exp(-gamma^2 + A306769).
%F A306765 Equals lim_{k->oo} k^(k*(2*k+1) + 2*gamma) * (2*Pi)^k * exp(1/6 + log(k)^2 - 2*k^2) / A306760(k).
%e A306765 2.0344489454876164779803555318869026355979439863702376260005284165650078277571...
%p A306765 evalf(exp(-gamma^2 + Sum((-1)^j*Zeta(j)^2/j, j=2..infinity)), 100);
%t A306765 slogam = Table[Sum[LogGamma[1/j], {j, 1, n}], {n, 1, 1000}]; $MaxExtraPrecision = 1000; funs[n_] := E^slogam[[n]] * n^EulerGamma/n!; Do[Print[N[Sum[(-1)^(m + j) * funs[j*Floor[Length[slogam]/m]] * (j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 80]], {m, 10, 100, 10}]
%o A306765 (PARI) exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))
%Y A306765 Cf. A001620, A231132, A303670, A306760, A306769.
%K A306765 nonn,cons
%O A306765 1,1
%A A306765 _Vaclav Kotesovec_, Mar 08 2019