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%I #28 Apr 04 2014 04:35:51
%S 1,-1,7,-43,4489,-49897,20130311,-319053131,329796121169,
%T -62717244921977,14635852695795623,-33233512260583073,
%U 149490010959849868177,-3562767949848393597053
%N Numerators of coefficients in function a(x) such that a(a(x)) = arctan(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
%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 a(n) = numerator(T(2*n-1,1)), T(n,m)=1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum(i=m..n, (2^i*stirling1(i,m)*binomial(n-1,i-1))/i!)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 12 2012
%e x - x^3/6 + x^5 * 7/120 + ...
%t n = 28; a[x_] = Sum[c[k] k! x^k, {k, 1, n, 2}];
%t sa = Series[a[x], {x, 0, n}];
%t coes = CoefficientList[ComposeSeries[sa, sa] - Series[ArcTan[x], {x, 0, n}], x] // Rest;
%t eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n, 2}] /. First[Solve[eq]] // Numerator
%t (* _Jean-François Alcover_, Apr 26 2011 *)
%o (Maxima)
%o T(n, m):=if n=m then 1 else 1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum((2^i*stirling1(i, m)*binomial(n-1, i-1))/i!, i, m, n)-sum(T(n, i)*T(i, m), i, m+1, n-1));
%o makelist(num(T(2*n-1, 1), n, 1, 5)); \\ _Vladimir Kruchinin_, Mar 12 2012
%Y Cf. A048604, A095885.
%K frac,sign,nice
%O 0,3
%A Winston C. Yang (yang(AT)math.wisc.edu)
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