
COMMENTS

There are two sixpattern 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)}.
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
