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%I #39 Dec 23 2021 02:41:40
%S 1,2,8,56,576,7872,134656,2771456,66744320,1842237440,57354338304,
%T 1988721131520,76015173369856,3175757373243392,143980934947930112,
%U 7040807787705663488,369414622819764928512,20700889684976244621312,1233951687316746828513280,77963762014950356953333760
%N Number of labeled rooted forests with 2-colored leaves.
%C a(n) is the number of labeled trees on vertices 0,1,...,n rooted at 0, where all leaves have 2 colors (except the singleton tree 0 has only 1 color).
%H Seiichi Manyama, <a href="/A349562/b349562.txt">Table of n, a(n) for n = 0..372</a>
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).
%F a(n) = A216857(n+1)/(n+1).
%F a(n) = A038049(n+1)/(n+1) for n>=1, and a(0) = A038049(1)/2.
%F a(n) = 2*A201595(n) - A000007(n).
%F E.g.f. satisfies: A(x) = e^(x*(1 + A(x))).
%F E.g.f. satisfies: A(-x*A(x)) = 1/A(x).
%F From _Vaclav Kotesovec_, Nov 25 2021: (Start)
%F E.g.f.: -LambertW(-x*exp(x))/x.
%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n+1)).
%F (End)
%F From _Seiichi Manyama_, Nov 26 2021: (Start)
%F G.f.: Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1).
%F a(n) = 2^n * A007889(n). (End)
%e a(2)=8 counts trees 0-1-2B, 0-1-2R, 0-2-1B, 0-2-1R, 1B-0-2B, 1B-0-2R, 1R-0-2B, 1R-0-2R (where B and R stand for colors Blue and Red).
%t CoefficientList[u/.AsymptoticSolve[u-E^(x(1+u))==0,u->1,{x,0,24}][[1]],x]Factorial/@Range[0,24]
%t nmax = 20; CoefficientList[Series[-LambertW[-x*Exp[x]]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 25 2021 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k+1)^(k-1)*x^k/(1-(k+1)*x)^(k+1))) \\ _Seiichi Manyama_, Nov 26 2021
%Y Cf. A000007, A007889, A216857, A038049, A201595,
%K nonn
%O 0,2
%A _Alexander Burstein_, Nov 22 2021
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