OFFSET
0,2
COMMENTS
A recursion exists for coefficients, but is too complicated to process without a computer algebra system.
REFERENCES
W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
W. C. Yang, Composition equations, preprint, 1999
LINKS
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x) = F(x), arXiv:1302.1986
W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.
EXAMPLE
x - x^3/6 + x^5 * 7/120 ...
MATHEMATICA
n = 28; a[x_] = Sum[c[k] k! x^k, {k, 1, n, 2}];
sa = Series[a[x], {x, 0, n}];
coes = CoefficientList[ComposeSeries[sa, sa] - Series[ArcTan[x], {x, 0, n}], x] // Rest;
eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n, 2}] /. First[Solve[eq]] // Denominator
(* Jean-François Alcover, Apr 26 2011 *)
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Winston C. Yang (yang(AT)math.wisc.edu)
STATUS
approved