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A048603 Denominators of coefficients in function a(x) such that a(a(x)) = sin x. 11
1, 12, 160, 40320, 71680, 1277337600, 79705866240, 167382319104000, 91055981592576000, 62282291409321984000, 4024394214140805120000, 5882770031248492462080000, 9076273762497674084352000000 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..12.

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

Cf. A048602, A048606.

Sequence in context: A144346 A167558 A048609 * A275040 A109391 A296194

Adjacent sequences:  A048600 A048601 A048602 * A048604 A048605 A048606

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

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Last modified December 4 17:50 EST 2021. Contains 349526 sequences. (Running on oeis4.)