OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
E.g.f.: exp( -LambertW(-x * log(1+x)) / x ) = (1+x)^exp( -LambertW(-x * log(1+x)) ).
E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (log(1+x))^k / k!.
a(n) ~ -(-1)^n * sqrt(1 + exp(1)*r^2/(1-r)) * n^(n-1) / (exp(n + 1/r) * r^(n+1)), where r = 0.5123112855238643734867005914814802444318611742265... is the root of the equation r*log(1-r) = -exp(-1). - Vaclav Kotesovec, Jan 30 2026
MATHEMATICA
nmax = 20; CoefficientList[Series[E^(-LambertW[-x*Log[1+x]]/x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 30 2026 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^exp(-lambertw(-x*log(1+x)))))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, May 04 2023
STATUS
approved