OFFSET
0,3
COMMENTS
Expansion of inverse of x*exp(2x).
Number of well-colored directed trees on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.
Number of labeled rooted directed trees on n nodes.
LINKS
Robert Israel, Table of n, a(n) for n = 0..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 702
FORMULA
E.g.f.: -1/2*W(-2*x), where W is Lambert's W function.
From Robert Israel, Jun 16 2016: (Start)
E.g.f. g(x) satisfies g(x) = x*exp(2*g(x)) and (1-2*g(x)) g'(x) = g(x).
a(n) = (2*n/(n-1)) * Sum_{j=1..n-1} binomial(n-1,j)*a(j)*a(n-j) for n >= 2. (End)
a(n) = [x^n] x/(1 - 2*n*x). - Ilya Gutkovskiy, Oct 12 2017
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - Stefano Spezia, Mar 12 2023
MAPLE
spec := [S, {B=Set(S), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
terms = 19;
A[x_] = -1/2 LambertW[-2 x];
CoefficientList[A[x] + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Jan 15 2019 *)
Join[{0}, Table[(2n)^(n-1), {n, 20}]] (* Harvey P. Dale, Dec 14 2020 *)
PROG
(Sage)[lucas_number1(n, 2*n, 0) for n in range(0, 17)] # Zerinvary Lajos, Mar 09 2009
(PARI) a(n)=if(n, (2*n)^(n-1), 0) \\ Charles R Greathouse IV, Nov 20 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
New description from Vladeta Jovovic, Mar 08 2003
STATUS
approved