1,2

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') - (1')^4 (6') ] * t^6 / 6!

...

See A134685 for more information.

The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [ (e(2))! * (e(3))! * ... * (e(n))! ] .

Sequence in context: A035877 A086494 A107414 this_sequence A014964 A001898 A002209

Adjacent sequences: A133434 A133435 A133436 this_sequence A133438 A133439 A133440

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Tom Copeland (tcjpn(AT)msn.com), Jan 27 2008