A097627 Number of rooted directed trees on n nodes with a red root.

1, 2, 21, 280, 5465, 134556, 4051453, 143810416, 5884797969, 272701388980, 14116335883661, 807328717090248, 50554260752606377, 3440140092356781100, 252777268861251990045, 19946622760623381708256

Offset: 1

Author

Ralf Stephan, Aug 17 2004

Keywords

• nonn

Comments

Ditrees are well-colored directed trees. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• C. Banderier, J.-M. Le Bars and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

Crossrefs

Cf. A097629.

Programs

Mathematica

Rest[CoefficientList[Series[LambertW[-LambertW[-2*x]/2], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 08 2013 *)

PARI

x='x+O('x^30); Vec(serlaplace(lambertw(-lambertw(-2*x)/2))) \\ G. C. Greubel, Nov 15 2017

Formula

a(n) = A052746(n) - A097628(n).

E.g.f.: -C(-C(2*x)/2), where C(x) is the e.g.f. of A000169.

a(n) ~ LambertW(1/2) * n^(n-1) * 2^n / (1+LambertW(1/2)). - Vaclav Kotesovec, Oct 08 2013