A048603 Denominators of coefficients in function a(x) such that a(a(x)) = sin x.

1, 12, 160, 40320, 71680, 1277337600, 79705866240, 167382319104000, 91055981592576000, 62282291409321984000, 4024394214140805120000, 5882770031248492462080000, 9076273762497674084352000000

Offset: 0

Author

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

Keywords

• frac
• nonn
• nice

Comments

Also denominators of coefficients in function a(x) such that a(a(x)) = sinh x.

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

Examples

			x - x^3/12 - x^5/160 ...
		

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. A048602, A048606.

Programs

Mathematica

n = 13; m = 2 n - 1 (* m = maximal degree *); a[x_] = Sum[c[k] x^k, {k, 1, m, 2}] ; coes = DeleteCases[
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}]
(CoefficientList[a[x] /. Flatten @ Array[s, n], x] // Denominator // Partition[#, 2] &)[[All, 2]]
(* Jean-François Alcover, May 05 2011 *)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007