A366271 - OEIS

%I #9 Oct 06 2023 10:53:01

%S 1,3,8,4,8,9,2,0,1,2,6,5,9,8,6,8,9,0,4,1,7,8,6,1,1,0,6,0,7,5,7,1,2,8,

%T 1,3,5,8,3,0,4,8,1,4,8,9,2,9,7,6,3,9,7,7,7,0,9,4,7,5,2,2,6,5,5,0,8,5,

%U 4,7,9,4,0,9,7,1,1,2,6,2,8,5,5,9,6,5,6,4,0,4,5,8,7,7,0,7,8,9,5,7,6,8,4,9,7

%N Decimal expansion of limit_{n->oo} Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n))^(1/n).

%C Limit_{n->oo} Product_{k=1..n} (k/n)^(k/n^2) = exp(-1/4).

%F Equals exp(-1/4 + Integral_{x=0..1} log(1 + (1/x - 1)^x) dx).

%F Conjecture: Limit_{n->oo} (1/A366271^n) * Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n)) = 1/sqrt(2).

%e 1.38489201265986890417861106075712813583048148929763977709475...

%t RealDigits[Exp@NIntegrate[Log[1+(1/r-1)^r], {r, 0, 1}, WorkingPrecision->120] * Exp[-1/4], 10, 105][[1]]

%Y Cf. A323575.

%K nonn,cons

%O 1,2

%A _Vaclav Kotesovec_, Oct 06 2023