A052746 a(0) = 0; a(n) = (2*n)^(n-1), n > 0.

0, 1, 4, 36, 512, 10000, 248832, 7529536, 268435456, 11019960576, 512000000000, 26559922791424, 1521681143169024, 95428956661682176, 6502111422497947648, 478296900000000000000, 37778931862957161709568, 3189059870763703892770816, 286511799958070431838109696

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

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

Crossrefs

Cf. A019762 (2*e), A038057, A097627.

Programs

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 *)

PARI

a(n)=if(n,(2*n)^(n-1),0) \\ Charles R Greathouse IV, Nov 20 2011

Sage

[lucas_number1(n, 2*n, 0) for n in range(0, 17)] # Zerinvary Lajos, Mar 09 2009

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

Extensions

New description from Vladeta Jovovic, Mar 08 2003