OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
E.g.f.: A(x) = exp( -LambertW(-x * sqrt(1+2*x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * 2^(n-k) * binomial(k/2,n-k)/k!.
From Vaclav Kotesovec, Apr 21 2024: (Start)
E.g.f.: -LambertW(-x*sqrt(1 + 2*x))/(x*sqrt(1 + 2*x)).
a(n) ~ sqrt(3*r + 1) * n^(n-1) / ((1 + 2*r)^(3/4) * exp(n - 1/2) * r^(n + 1/2)), where r = (exp(2/3) + (-exp(1) + (6*(9 + sqrt(81 - 3*exp(2))))/exp(1))^(2/3)) / (6*(54 - exp(2) + 6*sqrt(81 - 3*exp(2)))^(1/3)) - 1/6 = 0.292252770550601628... (End)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*sqrt(1+2*x)))))
(PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*2^(n-k)*binomial(k/2, n-k)/k!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 20 2024
STATUS
approved