OFFSET
0,2
COMMENTS
Inverse binomial transform of A027710.
In general, if m >= 1 and e.g.f. = exp(m*exp(x) + r*x + s) then
a(n) ~ n^(n+r) * exp(n/LambertW(n/m) - n + s) / (m^r * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n+r)).
Equivalently, a(n) ~ n! * (n/m)^r * exp(n/LambertW(n/m) + s) / (sqrt(2*Pi*n * (1 + LambertW(n/m))) * LambertW(n/m)^(n+r)).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..545
FORMULA
a(n) ~ 3 * n^(n-1) * exp(n/LambertW(n/3) - n - 3) / (sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n-1)).
a(0) = 1; a(n) = -a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023
MATHEMATICA
nmax = 25; CoefficientList[Series[Exp[3*Exp[x]-3-x], {x, 0, nmax}], x] * Range[0, nmax]!
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - x))) \\ Michel Marcus, Dec 04 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jun 26 2022
STATUS
approved