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A048602
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Numerators of coefficients in function a(x) such that a(a(x)) = sin x.
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11
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1, -1, -1, -53, -23, -92713, -742031, 594673187, 329366540401, 104491760828591, 1508486324285153, -582710832978168221, -1084662989735717135537, -431265609837882130202597, 784759327625761394688977441
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OFFSET
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0,4
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COMMENTS
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Recursion exists for coefficients, but is too complicated to process without computer algebra system
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REFERENCES
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W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
W. C. Yang, Composition equations, preprint, 1999
W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.
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LINKS
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Table of n, a(n) for n=0..14.
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986
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FORMULA
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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)). [From Vladimir Kruchinin, Nov 08 2011]
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EXAMPLE
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x - x^3/12 - x^5/160 ...
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MATHEMATICA
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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]] (* From Jean-François Alcover, May 05 2011 *)
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PROG
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(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); [From Vladimir Kruchinin, Nov 08 2011]
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CROSSREFS
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Cf. A048603. Apart from signs, the same sequence as A048606.
Sequence in context: A143428 A143385 * A048606 A033373 A171132 A104936
Adjacent sequences: A048599 A048600 A048601 * A048603 A048604 A048605
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KEYWORD
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frac,sign,nice
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AUTHOR
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Winston C. Yang (yang(AT)math.wisc.edu)
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STATUS
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approved
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