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First few terms coincide with A032120 but A032120(8) = 3321. This corresponds to the fact that A032120 gives dimensions of components of the free special Jordan algebra (which follows from Cohn 1959), and 3324 - 3321 = 3 is the dimension of the GL_3-orbit of the so called Glennie identity.

The terms up to a(12) were computed using the Albert nonassociative algebra system.

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)}.

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

Name was: Dimensions of components for the operad of level algebras.