OFFSET
0,2
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.
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^3/12 - x^5/160 ...
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 *)
CROSSREFS
KEYWORD
frac,nonn,nice
AUTHOR
Winston C. Yang (yang(AT)math.wisc.edu)
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007
STATUS
approved