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A052757 Number of rooted identity trees with n nodes and 3-colored non-root nodes.

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