A097630 Number of unrooted directed trees on n nodes with a red root.

1, 1, 7, 70, 1093, 22426, 578779, 17976302, 653866441, 27270138898, 1283303262151, 67277393090854, 3888789288662029, 245724292311198650, 16851817924083466003, 1246663922538961356766, 98960637955717312632721

Offset: 1

Author

Ralf Stephan, Aug 17 2004

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• C. Banderier, J.-M. Le Bars and V. Ravelomanana, Generating functions for kernels of digraphs

Crossrefs

Cf. A097629.

Programs

Mathematica

terms = 17;
Rest[CoefficientList[LambertW[-LambertW[-2x]/2] + O[x]^(terms+1), x]]* Range[0, terms-1]! (* Jean-François Alcover, Dec 13 2018 *)

PARI

seq(n)={Vec(serlaplace(lambertw(-lambertw(-2*x + O(x*x^n))/2)/x))} \\ Andrew Howroyd, Dec 13 2018

Formula

E.g.f.: A(x) = 2B - 2BC + C - 2B/C + C^2/2, with B = egf of A052746 and C = egf of A097627.

a(n) = (n-1)!*[x^n](LambertW(-LambertW(-2x)/2)). - Jean-François Alcover, Dec 13 2018

a(n) ~ 2^n * n^(n-2) / (1 + 1/LambertW(1/2)). - Vaclav Kotesovec, Feb 24 2019