A097172 Total number of brown nodes among tricolored labeled trees on n nodes.

3, 4, 185, 1026, 30457, 362664, 10245825, 195060070, 5907674201, 153676400076, 5199628119985, 169205814335754, 6462995557999905, 249877775352089296, 10749867848389013249, 478345428286978038606, 23013713995857481324969

Offset: 3

Author

Ralf Stephan, Jul 30 2004

Keywords

• nonn

Links

• S. Coulomb and M. Bauer, On vertex covers, matchings and random trees

Crossrefs

Equals A000169(n)-A097174(n)-A097173(n).

Cf. A097170, A097171, A097173, A097174, A000272.

Programs

Mathematica

Drop[CoefficientList[Series[-LambertW[-x] - LambertW[-LambertW[-x]]- LambertW[-LambertW[-x]]^2, {x, 0, 20}], x] * Range[0, 20]!, 3] (* Vaclav Kotesovec, Aug 26 2016 *)

Formula

E.g.f.: A(x) = T(x)+T(-T(x))-T(-T(x))^2, with T(x)=Sum[k=1..inf, A000169(k)/k!*x^k].

a(n) = -n^(n-1) * {1 + Sum[l=1..n, (-l/n)^l*(2/l-1)*C(n, l)]}.

a(n) ~ (1-2*LambertW(1)^2)*n^(n-1)/(1+LambertW(1)). - Vaclav Kotesovec, Aug 26 2016