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%I #7 Aug 26 2016 02:44:04
%S 3,4,185,1026,30457,362664,10245825,195060070,5907674201,153676400076,
%T 5199628119985,169205814335754,6462995557999905,249877775352089296,
%U 10749867848389013249,478345428286978038606,23013713995857481324969
%N Total number of brown nodes among tricolored labeled trees on n nodes.
%H S. Coulomb and M. Bauer, <a href="http://arXiv.org/abs/math.CO/0407456">On vertex covers, matchings and random trees</a>
%F 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].
%F a(n) = -n^(n-1) * {1 + Sum[l=1..n, (-l/n)^l*(2/l-1)*C(n, l)]}.
%F a(n) ~ (1-2*LambertW(1)^2)*n^(n-1)/(1+LambertW(1)). - _Vaclav Kotesovec_, Aug 26 2016
%t 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 *)
%Y Equals A000169(n)-A097174(n)-A097173(n).
%Y Cf. A097170, A097171, A097173, A097174, A000272.
%K nonn
%O 3,1
%A _Ralf Stephan_, Jul 30 2004