A048607 Numerators of coefficients in function a(x) such that a(a(x)) = log(1+x).

1, -1, 5, -5, 109, -497, 127, -11569, 312757, -1219255, 165677473, -885730939, 20163875141, -252312616027, 9565074633871, -691138954263097, 5061676927076641, -95993669516238563, 1245671625068799013

Offset

0,3

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..18.

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

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

Formula

T(n,m) = if n=m then 1 else (stirling1(n,m)*m!/n!-sum(i=m+1..n-1, T(n,i)*T(i,m)))/2; a(n)=T(n,1), n>0. - Vladimir Kruchinin, Nov 08 2011

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}]]] // Numerator (* 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] // Numerator; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 16 2014, after Vladimir Kruchinin *)

Crossrefs

Cf. A048608.

Sequence in context: A151492 A081050 A081049 * A229767 A369753 A215729

Adjacent sequences: A048604 A048605 A048606 * A048608 A048609 A048610

Keyword

frac,sign,nice

Author

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

Status

approved