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Denominators of coefficients in function a(x) such that a(a(x)) = arctan(x).
1

%I #26 Apr 04 2014 04:36:04

%S 1,6,120,1680,362880,7983360,6227020800,186810624000,355687428096000,

%T 121645100408832000,51090942171709440000,213653030899875840000,

%U 1723467782592331776000000,64431180179990249472000000

%N Denominators of coefficients in function a(x) such that a(a(x)) = arctan(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^3/6 + 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]] // Denominator

%t (* _Jean-François Alcover_, Apr 26 2011 *)

%Y Cf. A048605.

%K frac,nonn

%O 0,2

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