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

0,2

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

Table of n, a(n) for n=0..13.

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

Cf. A048605.

Sequence in context: A351862 A170917 A115678 * A001516 A350712 A026337

Adjacent sequences: A048601 A048602 A048603 * A048605 A048606 A048607

Keyword

frac,nonn

Author

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

Status

approved