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

%I #37 Feb 13 2016 06:34:04

%S 1,-1,5,-5,109,-497,127,-11569,312757,-1219255,165677473,-885730939,

%T 20163875141,-252312616027,9565074633871,-691138954263097,

%U 5061676927076641,-95993669516238563,1245671625068799013

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

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

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

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

%H Dmitry Kruchinin, Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation A^{2^n}(x)=F(x)</a>, arXiv:1302.1986 [math.CO], 2013.

%H W. C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Math., 222 (2000), 235-245.

%F 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

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

%t 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 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_ *)

%Y Cf. A048608.

%K frac,sign,nice

%O 0,3

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