A048604 Denominators of coefficients in function a(x) such that a(a(x)) = arctan(x).

1, 6, 120, 1680, 362880, 7983360, 6227020800, 186810624000, 355687428096000, 121645100408832000, 51090942171709440000, 213653030899875840000, 1723467782592331776000000, 64431180179990249472000000

Offset: 0

Author

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

Keywords

• frac
• nonn

Comments

A recursion exists for coefficients, but is too complicated to process without a computer algebra system.

Examples

			x - x^3/6 + x^5 * 7/120 ...
		

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.

Crossrefs

Cf. A048605.

Programs

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 *)