A048608 Denominators of coefficients in function a(x) such that a(a(x)) = log(1+x).

1, 4, 48, 96, 3840, 30720, 13440, 2064384, 92897280, 594542592, 130799370240, 1121137459200, 40809403514880, 816188070297600, 48971284217856000, 5484783832399872000, 62160883433865216000, 1918107260244983808000

Offset

0,2

Comments

A recursion exists for coefficients, but is too complicated to use without a computer algebra system.

References

W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999

W. C. Yang, Composition equations, preprint, 1999

Links

Table of n, a(n) for n=0..17.

W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.

Example

x - x^2/4 + x^3 * 5/48 + ...

Mathematica

n = 18; a[x_] = Sum[c[k] k! x^k, {k, 1, n}]; sa = Series[a[x], {x, 0, n}]; coes = CoefficientList[ ComposeSeries[sa, sa] - Series[Log[1+x], {x, 0, n}], x] // Rest; eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n}] /. First[Solve[eq, Table[c[k], {k, 1, n}]]] // Denominator (* Jean-François Alcover, Mar 28 2011  + upgrading by Olivier Gérard *)
T[n_, m_] := T[n, m] = If[n == m, 1, (StirlingS1[n, m]*m!/n! - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])/2]; a[n_] := T[n+1, 1] // Denominator; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 16 2014, after Vladimir Kruchinin *)

Crossrefs

Cf. A048607.

Sequence in context: A157818 A362402 A373435 * A366492 A275033 A192418

Adjacent sequences: A048605 A048606 A048607 * A048609 A048610 A048611

Keyword

frac,nonn,nice

Author

Winston C. Yang (yang(AT)math.wisc.edu)

Status

approved