A366271 Decimal expansion of limit_{n->oo} Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n))^(1/n).

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, 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, 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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..105.

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

Cf. A323575.

Sequence in context: A340782 A010628 A225456 * A212886 A127438 A303217

Adjacent sequences: A366268 A366269 A366270 * A366272 A366273 A366274

Keyword

nonn,cons

Author

Vaclav Kotesovec, Oct 06 2023

Status

approved