OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = n! * Sum_{k=0..floor(n/4)} (n-3*k+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.
E.g.f.: exp( -LambertW(-x*(1+x^3)) ).
From Vaclav Kotesovec, Oct 10 2024: (Start)
E.g.f.: -LambertW(-x*(1+x^3))/(x*(1+x^3)).
a(n) ~ sqrt(1 + 4*r^3) * n^(n-1) / (exp(n - 3/2) * r^(n - 1/2)), where r = 0.3524487947753533587882657130849651130602232209406... is the positive real root of the equation r*(1 + r^3) = exp(-1). (End)
PROG
(PARI) a(n) = n!*sum(k=0, n\4, (n-3*k+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x^3)))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 28 2024
STATUS
approved