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%I #26 Apr 13 2019 22:19:26
%S 0,1,3,12,64,363,2214,14043,91857,614676,4189254,28974915,202870938,
%T 1435094800,10241197917,73639001172,533004547453,3880381334415,
%U 28395656513145,208748382089131,1540935621796941,11417266889312313,84880193073070819,632976019285857201
%N Number of rooted identity trees with n nodes and 3-colored non-root nodes.
%C Previous name was: A simple grammar.
%H Alois P. Heinz, <a href="/A052757/b052757.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=713">Encyclopedia of Combinatorial Structures 713</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938424..., c = 0.12982932099206082951153936270704832022771078... . - _Vaclav Kotesovec_, Feb 24 2015
%F From _Ilya Gutkovskiy_, Apr 13 2019: (Start)
%F G.f. A(x) satisfies: A(x) = x*exp(3*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(3*a(n)). (End)
%e a(3) = 12:
%e o o o o o o o o o o o o
%e | | | | | | | | | / \ / \ / \
%e 1 1 1 2 2 2 3 3 3 1 2 1 3 2 3
%e | | | | | | | | |
%e 1 2 3 1 2 3 1 2 3 - _Alois P. Heinz_, Feb 24 2015
%p spec := [S,{S=Prod(B,B,B,Z),B=PowerSet(S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%Y Cf. A038079.
%Y Column k=3 of A255517.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name from _Vaclav Kotesovec_, Feb 24 2015
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