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%I #54 Feb 20 2022 23:00:14
%S 1,-1,-1,-53,-23,-92713,-742031,594673187,329366540401,
%T 104491760828591,1508486324285153,-582710832978168221,
%U -1084662989735717135537,-431265609837882130202597,784759327625761394688977441
%N Numerators of coefficients in function a(x) such that a(a(x)) = sin(x).
%C A recursion exists for coefficients, but is too complicated to process 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 and 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 ((((-1)^(n-m)+1)*sum(i=0..m/2, (2*i-m)^n *binomial(m,i)*(-1)^((n+m)/2-i)))/(2^m*n!) -sum(T(n,i)*T(i,m), i=m+1..n-1))/2; a(n)=numerator(T(n,1)). - _Vladimir Kruchinin_, Nov 08 2011
%F a(n) = numerator( A098932(n)/(2^(n-1) * (2*n-1)!) ). - _Andrew Howroyd_, Feb 20 2022
%e x - x^3/12 - x^5/160 ...
%t n = 15; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[CoefficientList[Series[a@a@x - Sin[x], {x, 0, m}], x] // Rest , 0]; Do[s[k] = Solve[coes[[1]] == 0] // First; coes = coes /. s[k] // Rest, {k, 1, n}]; (- CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Numerator // Partition[#, 2] &)[[All, 2]] (* _Jean-François Alcover_, May 05 2011 *)
%o (Maxima)
%o T(n,m):= if n=m then 1 else ((((-1)^(n-m)+1)*sum((2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i),i,0,m/2))/(2^m*n!)-sum(T(n,i)*T(i,m),i,m+1,n-1))/2; makelist(num(T(n,1)),n,1,10); \\ _Vladimir Kruchinin_, Nov 08 2011
%o (PARI) a(n) = { my(ps = sin(x + O(x^(2*n))), q=0); while(ps<>q, q=ps; ps=(sin(serreverse(ps)) + ps)/2); numerator(polcoef(ps, 2*n-1)) } \\ _Gottfried Helms_, Feb 20 2022
%Y Denominators are A048603.
%Y Apart from signs, the same sequence as A048606.
%Y Cf. A098932 (normalized version).
%K frac,sign,nice
%O 0,4
%A Winston C. Yang (yang(AT)math.wisc.edu)
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