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%I #20 Jun 16 2015 08:19:55
%S 1,2,6,42,390,4698,69174
%N Number of binary labeled trees with two-colored vertices that have n leaves and avoid the easiest to avoid 6-pattern set.
%C There are two six-pattern sets that are the easiest to avoid, they are identified with one another by either swapping colors (black <-> white) or passing to complements (the latter implies that the compositional inverse e.g.f. F(x) of the sequence in question is -F(-x)). One of them is (in operation notation, with b/w encoding black/white vertices) {b(b(1,2),3), b(b(1,3),2), b(1,b(2,3)), b(w(1,3),2), b(1,w(2,3)), w(b(1,2),3)}, the other is {w(w(1,2),3), w(w(1,3),2), w(1,w(2,3)), w(b(1,3),2), w(1,b(2,3)), b(w(1,2),3)}.
%C Conjecture: E.g.f. (for offset 0) satisfies A'(x) = 1 + A(x)^3, with A(0)=1. The next terms are 1203498, 24163110, 549811962, 13982486166, 393026414922, ... - _Vaclav Kotesovec_, Jun 15 2015
%D V. Dotsenko, Pattern avoidance in labelled trees, Séminaire Lotharingien de Combinatoire, B67b (2012), 27 pp.
%Y Cf. A258880, A258969.
%K nonn,more
%O 1,2
%A _Vladimir Dotsenko_, Jul 04 2013
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