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%I #22 Sep 17 2022 14:29:31
%S 1,0,6,12,320,2190,51492,685496,17286768,348213690,9956411300,
%T 266065478052,8737396913544,287741445880070,10816320294520860,
%U 420123621828718320,17913098835916877792,798053882730994171890,38192029991097097185108,1914946396460982552420380
%N Total number of red nodes among tricolored labeled trees on n nodes.
%H G. C. Greubel, <a href="/A097174/b097174.txt">Table of n, a(n) for n = 1..385</a>
%H S. Coulomb and M. Bauer, <a href="https://arxiv.org/abs/math/0407456">On vertex covers, matchings and random trees</a>, arXiv:math/0407456 [math.CO], 2004.
%F E.g.f.: A(x) = -T(-T(x)), with T(x) = Sum_{k>=1} A000169(k)/k!*x^k.
%F a(n) = -n^(n-1) * Sum_{j=1..n} (-j/n)^j*C(n, j).
%F a(n) ~ LambertW(1)*n^(n-1)/(1+LambertW(1)). - _Vaclav Kotesovec_, Aug 26 2016
%t Rest[CoefficientList[Series[LambertW[-LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Aug 26 2016 *)
%o (PARI) x='x+O('x^50); Vec(serlaplace(lambertw(-lambertw(-x)))) \\ _G. C. Greubel_, Nov 15 2017
%Y Cf. A097170, A097171, A097172, A097173, A000272, A000169.
%K nonn
%O 1,3
%A _Ralf Stephan_, Jul 30 2004
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