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 A048607 Numerators of coefficients in function a(x) such that a(a(x)) = log(1+x). 1
 1, -1, 5, -5, 109, -497, 127, -11569, 312757, -1219255, 165677473, -885730939, 20163875141, -252312616027, 9565074633871, -691138954263097, 5061676927076641, -95993669516238563, 1245671625068799013 (list; graph; refs; listen; history; text; internal format)
 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 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 A215729 A094463 Adjacent sequences:  A048604 A048605 A048606 * A048608 A048609 A048610 KEYWORD frac,sign,nice AUTHOR Winston C. Yang (yang(AT)math.wisc.edu) STATUS approved

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Last modified January 20 17:05 EST 2019. Contains 319335 sequences. (Running on oeis4.)