A097627 - OEIS

A097627

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

%I #20 Nov 16 2017 02:55:29

%S 1,2,21,280,5465,134556,4051453,143810416,5884797969,272701388980,

%T 14116335883661,807328717090248,50554260752606377,3440140092356781100,

%U 252777268861251990045,19946622760623381708256

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

%C 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.

%H Vincenzo Librandi, <a href="/A097627/b097627.txt">Table of n, a(n) for n = 1..100</a>

%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.

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

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

%F a(n) ~ LambertW(1/2) * n^(n-1) * 2^n / (1+LambertW(1/2)). - _Vaclav Kotesovec_, Oct 08 2013

%t Rest[CoefficientList[Series[LambertW[-LambertW[-2*x]/2], {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 08 2013 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(lambertw(-lambertw(-2*x)/2))) \\ _G. C. Greubel_, Nov 15 2017

%Y Cf. A097629.

%K nonn

%O 1,2

%A _Ralf Stephan_, Aug 17 2004