A052746 - OEIS

%I #53 Aug 15 2023 17:01:51

%S 0,1,4,36,512,10000,248832,7529536,268435456,11019960576,512000000000,

%T 26559922791424,1521681143169024,95428956661682176,

%U 6502111422497947648,478296900000000000000,37778931862957161709568,3189059870763703892770816,286511799958070431838109696

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

%C Expansion of inverse of x*exp(2x).

%C 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.

%C Number of labeled rooted directed trees on n nodes.

%H Robert Israel, <a href="/A052746/b052746.txt">Table of n, a(n) for n = 0..350</a>

%H Federico Ardila, Matthias Beck, and Jodi McWhirter, <a href="https://arxiv.org/abs/2004.02952">The Arithmetic of Coxeter Permutahedra</a>, arXiv:2004.02952 [math.CO], 2020.

%H C. Banderier, J.-M. Le Bars, and V. Ravelomanana, <a href="https://arxiv.org/abs/math/0411138">Generating functions for kernels of digraphs</a>, arXiv:math/0411138 [math.CO], 2004.

%H Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, <a href="https://arxiv.org/abs/2308.04751">Hurwitz numbers for reflection groups III: Uniform formulas</a>, arXiv:2308.04751 [math.CO], 2023, see p. 11.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=702">Encyclopedia of Combinatorial Structures 702</a>

%F E.g.f.: -1/2*W(-2*x), where W is Lambert's W function.

%F From _Robert Israel_, Jun 16 2016: (Start)

%F E.g.f. g(x) satisfies g(x) = x*exp(2*g(x)) and (1-2*g(x)) g'(x) = g(x).

%F a(n) = (2*n/(n-1)) * Sum_{j=1..n-1} binomial(n-1,j)*a(j)*a(n-j) for n >= 2. (End)

%F a(n) = [x^n] x/(1 - 2*n*x). - _Ilya Gutkovskiy_, Oct 12 2017

%F Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - _Stefano Spezia_, Mar 12 2023

%p spec := [S,{B=Set(S),S=Prod(Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t terms = 19;

%t A[x_] = -1/2 LambertW[-2 x];

%t CoefficientList[A[x] + O[x]^terms, x] Range[0, terms-1]! (* _Jean-François Alcover_, Jan 15 2019 *)

%t Join[{0},Table[(2n)^(n-1),{n,20}]] (* _Harvey P. Dale_, Dec 14 2020 *)

%o (Sage)[lucas_number1(n, 2*n, 0) for n in range(0, 17)] # _Zerinvary Lajos_, Mar 09 2009

%o (PARI) a(n)=if(n,(2*n)^(n-1),0) \\ _Charles R Greathouse IV_, Nov 20 2011

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

%K easy,nonn

%O 0,3

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

%E New description from _Vladeta Jovovic_, Mar 08 2003