1,3

Let f(t) = -ln(1 - u(.)*t) = sum(n=1,2,...) (u_n / 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) * [ -1 (2') ] * t^2 / 2!

P(3,t) = (1')^(-5) * [ 3 (2')^2 - 2 (1')(3') ] * t^3 / 3!

P(4,t) = (1')^(-7) * [ -15 (2')^3 + 20 (1')(2')(3') - 6 (1')^2 (4') ] * t^4 / 4!

P(5,t) = (1')^(-9) * [ 105 (2')^4 - 210 (1') (2')^2 (3') + 90 (1')^2 (2') (4') + 40 (1')^2 (3')^2 - 24 (1')^3 (5') ] * t^5 / 5!

P(6,t) = (1')^(-11) * [ -945 (2')^5 + 2520 (1') (2')^3 (3') - 1120 (1')^2 (2') (3')^2 - 1260 (1')^2 (2')^2 (4') + 504 (1')^3 (2')(5') + 420 (1')^3 (3')(4') - 120 (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)]! / [ 2^e(2) (e(2))! * 3^e(3) (e(3))! * ... n^e(n) * (e(n))! ] .

Adjacent sequences: A133929 A133930 A133931 this_sequence A133933 A133934 A133935

Sequence in context: A103236 A141235 A051917 this_sequence A111999 A126323 A084886

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