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%I #30 Dec 23 2021 03:29:55
%S 2,4,24,224,2880,47232,942592,22171648,600698880,18422374400,
%T 630897721344,23864653578240,988197253808128,44460603225407488,
%U 2159714024218951680,112652924603290615808,6280048587936003784704,372616014329572403183616,23445082059018189741752320
%N Number of labeled rooted trees with 2-colored leaves.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.83)
%H Alois P. Heinz, <a href="/A038049/b038049.txt">Table of n, a(n) for n = 1..150</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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Divides by n and shifts left under exponential transform.
%F E.g.f.: A(x) = x-LambertW(-x*exp(x)). - _Vladeta Jovovic_, Mar 08 2003
%F a(n) = Sum_{k=0..n} (binomial(n, k)*(n-k)^(n-1)).
%F A(x) = 2*compositional inverse of 2*x/(1+exp(2*x)). - _Peter Bala_, Oct 14 2011
%F a(n) ~ n^(n-1) * sqrt((1+LambertW(1/e))) / (e*LambertW(1/e))^n. - _Vaclav Kotesovec_, Nov 30 2012
%p a:= n-> add(binomial(n, k)*(n-k)^(n-1), k=0..n):
%p seq(a(n), n=1..20); # _Alois P. Heinz_, Nov 30 2012
%t Table[n!*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,20}] (* _Vaclav Kotesovec_, Nov 30 2012 *)
%Y Cf. A000169, A029856, A038050, A038054, A088789.
%K nonn,eigen
%O 1,1
%A _Christian G. Bower_, Jan 04 1999
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