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%I #27 Sep 24 2018 16:53:13
%S 1,12,160,40320,71680,1277337600,79705866240,167382319104000,
%T 91055981592576000,62282291409321984000,4024394214140805120000,
%U 5882770031248492462080000,9076273762497674084352000000
%N Denominators of coefficients in function a(x) such that a(a(x)) = sin x.
%C Also denominators of coefficients in function a(x) such that a(a(x)) = sinh 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, 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.
%e x - x^3/12 - x^5/160 ...
%t n = 13; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[
%t 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}]
%t (CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Denominator // Partition[#, 2] &)[[All, 2]]
%t (* _Jean-François Alcover_, May 05 2011 *)
%Y Cf. A048602, A048606.
%K frac,nonn,nice
%O 0,2
%A Winston C. Yang (yang(AT)math.wisc.edu)
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 15 2007
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