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A048607
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Numerators of coefficients in function a(x) such that a(a(x)) = log(1+x).
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1
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1, -1, 5, -5, 109, -497, 127, -11569, 312757, -1219255, 165677473, -885730939, 20163875141, -252312616027, 9565074633871, -691138954263097, 5061676927076641, -95993669516238563, 1245671625068799013
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OFFSET
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0,3
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COMMENTS
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A recursion exists for coefficients, but is too complicated to use without a computer algebra system.
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REFERENCES
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W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
W. C. Yang, Composition equations, preprint, 1999
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LINKS
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FORMULA
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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
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EXAMPLE
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x - x^2/4 + x^3 * 5/48 + ...
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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frac,sign,nice
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AUTHOR
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Winston C. Yang (yang(AT)math.wisc.edu)
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STATUS
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approved
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