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A048605
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Numerators of coefficients in function a(x) such that a(a(x)) = arctan(x).
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2
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1, -1, 7, -43, 4489, -49897, 20130311, -319053131, 329796121169, -62717244921977, 14635852695795623, -33233512260583073, 149490010959849868177, -3562767949848393597053
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OFFSET
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0,3
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COMMENTS
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A recursion exists for coefficients, but is too complicated to use without a 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
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LINKS
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FORMULA
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a(n) = numerator(T(2*n-1,1)), T(n,m)=1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum(i=m..n, (2^i*stirling1(i,m)*binomial(n-1,i-1))/i!)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 12 2012
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EXAMPLE
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x - x^3/6 + x^5 * 7/120 + ...
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MATHEMATICA
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n = 28; a[x_] = Sum[c[k] k! x^k, {k, 1, n, 2}];
sa = Series[a[x], {x, 0, n}];
coes = CoefficientList[ComposeSeries[sa, sa] - Series[ArcTan[x], {x, 0, n}], x] // Rest;
eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n, 2}] /. First[Solve[eq]] // Numerator
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PROG
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(Maxima)
T(n, m):=if n=m then 1 else 1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum((2^i*stirling1(i, m)*binomial(n-1, i-1))/i!, i, m, n)-sum(T(n, i)*T(i, m), i, m+1, n-1));
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CROSSREFS
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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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