A097174 Total number of red nodes among tricolored labeled trees on n nodes.

1, 0, 6, 12, 320, 2190, 51492, 685496, 17286768, 348213690, 9956411300, 266065478052, 8737396913544, 287741445880070, 10816320294520860, 420123621828718320, 17913098835916877792, 798053882730994171890, 38192029991097097185108, 1914946396460982552420380

Offset: 1

Author

Ralf Stephan, Jul 30 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..385
• S. Coulomb and M. Bauer, On vertex covers, matchings and random trees, arXiv:math/0407456 [math.CO], 2004.

Crossrefs

Cf. A097170, A097171, A097172, A097173, A000272, A000169.

Programs

Mathematica

Rest[CoefficientList[Series[LambertW[-LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 26 2016 *)

PARI

x='x+O('x^50); Vec(serlaplace(lambertw(-lambertw(-x)))) \\ G. C. Greubel, Nov 15 2017

Formula

E.g.f.: A(x) = -T(-T(x)), with T(x) = Sum_{k>=1} A000169(k)/k!*x^k.

a(n) = -n^(n-1) * Sum_{j=1..n} (-j/n)^j*C(n, j).

a(n) ~ LambertW(1)*n^(n-1)/(1+LambertW(1)). - Vaclav Kotesovec, Aug 26 2016