OFFSET
1,2
COMMENTS
Limit_{n->oo} Product_{k=1..n} (k/n)^(k/n^2) = exp(-1/4).
FORMULA
Equals exp(-1/4 + Integral_{x=0..1} log(1 + (1/x - 1)^x) dx).
Conjecture: Limit_{n->oo} (1/A366271^n) * Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n)) = 1/sqrt(2).
EXAMPLE
1.38489201265986890417861106075712813583048148929763977709475...
MATHEMATICA
RealDigits[Exp@NIntegrate[Log[1+(1/r-1)^r], {r, 0, 1}, WorkingPrecision->120] * Exp[-1/4], 10, 105][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Oct 06 2023
STATUS
approved