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 A048602 Numerators of coefficients in function a(x) such that a(a(x)) = sin(x). 11
 1, -1, -1, -53, -23, -92713, -742031, 594673187, 329366540401, 104491760828591, 1508486324285153, -582710832978168221, -1084662989735717135537, -431265609837882130202597, 784759327625761394688977441 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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 Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x) = F(x), arXiv:1302.1986 [math.CO], 2013. W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245. FORMULA T(n,m) = if n=m then 1 else ((((-1)^(n-m)+1)*sum(i=0..m/2, (2*i-m)^n *binomial(m,i)*(-1)^((n+m)/2-i)))/(2^m*n!) -sum(T(n,i)*T(i,m), i=m+1..n-1))/2; a(n)=numerator(T(n,1)). - Vladimir Kruchinin, Nov 08 2011 EXAMPLE x - x^3/12 - x^5/160 ... MATHEMATICA n = 15; 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] // Numerator // Partition[#, 2] &)[[All, 2]] (* Jean-François Alcover, May 05 2011 *) PROG (Maxima) T(n, m):= if n=m then 1 else ((((-1)^(n-m)+1)*sum((2*i-m)^n*binomial(m, i)*(-1)^((n+m)/2-i), i, 0, m/2))/(2^m*n!)-sum(T(n, i)*T(i, m), i, m+1, n-1))/2; makelist(num(T(n, 1)), n, 1, 10); \\ Vladimir Kruchinin, Nov 08 2011 CROSSREFS Cf. A048603. Apart from signs, the same sequence as A048606. Sequence in context: A297984 A298633 A298710 * A048606 A033373 A289237 Adjacent sequences:  A048599 A048600 A048601 * A048603 A048604 A048605 KEYWORD frac,sign,nice AUTHOR Winston C. Yang (yang(AT)math.wisc.edu) STATUS approved

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Last modified December 2 17:52 EST 2020. Contains 338880 sequences. (Running on oeis4.)