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Let f(t) = u(t) - u(0) = sum(n=1,2,...) u_n * t^n .
If u_1 is not equal to 0, then the compositional inverse for f(t) is given by g(t) = sum(j=1,...) P(n,t) where, with u_n denoted by (n'),
P(1,t) = (1')^(-1) * [ 1 ] * t
P(2,t) = (1')^(-3) * [ -2 (2') ] * t^2 / 2!
P(3,t) = (1')^(-5) * [ 12 (2')^2 - 6 (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(-7) * [ -120 (2')^3 + 120 (1')(2')(3') - 24 (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(-9) * [ 1680 (2')^4 - 2520 (1') (2')^2 (3') + 720 (1')^2 (2') (4') + 360 (1')^2 (3')^2 - 120 (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(-11) * [ -30240 (2')^5 + 60480 (1') (2')^3 (3') - 20160 (1')^2 (2') (3')^2 - 20160 (1')^2 (2')^2 (4') + 5040 (1')^3 (2')(5') + 5040 (1')^3 (3')(4') - 720 (1')^4 (6') ] * t^6 / 6!
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See A134685 for more information.
A111785 is obtained from A133437 by dividing through the bracketed terms of the P(n,t) by n! and unsigned A111785 appears to be a refinement of A033282 and A126216. [From Tom Copeland (tcjpn(AT)msn.com), Sep 28 2008]
For relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see the Loday and McCammond links. E.g., P(5,t) = (1')^(-9) * [ 14 (2')^4 - 21 (1') (2')^2 (3') + 6 (1')^2 (2') (4')+ 3 (1')^2 (3')^2 - 1 (1')^3 (5') ] * t^5 is related to the 3-D associahedron with 14 vertices (0-D faces), 21 edges (1-D faces), 6 pentagons (2-D faces), 3 rectangles (2-D faces), 1 3-D polytope (3-D faces). Summing over faces of the same dimension gives A033282 or A126216. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]
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