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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 and 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
a(n) = numerator( A098932(n)/(2^(n-1) * (2*n-1)!) ). - Andrew Howroyd, Feb 20 2022
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
(PARI) a(n) = { my(ps = sin(x + O(x^(2*n))), q=0); while(ps<>q, q=ps; ps=(sin(serreverse(ps)) + ps)/2); numerator(polcoef(ps, 2*n-1)) } \\ Gottfried Helms, Feb 20 2022
CROSSREFS
Denominators are A048603.
Apart from signs, the same sequence as A048606.
Cf. A098932 (normalized version).
Sequence in context: A297984 A298633 A298710 * A048606 A033373 A289237
KEYWORD
frac,sign,nice
AUTHOR
Winston C. Yang (yang(AT)math.wisc.edu)
STATUS
approved

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Last modified April 22 12:06 EDT 2024. Contains 371900 sequences. (Running on oeis4.)