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 A048605 Numerators of coefficients in function a(x) such that a(a(x)) = arctan(x). 2
 1, -1, 7, -43, 4489, -49897, 20130311, -319053131, 329796121169, -62717244921977, 14635852695795623, -33233512260583073, 149490010959849868177, -3562767949848393597053 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A recursion exists for coefficients, but is too complicated to use 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 W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245. FORMULA 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 EXAMPLE x - x^3/6 + x^5 * 7/120 + ... MATHEMATICA 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 (* Jean-François Alcover, Apr 26 2011 *) PROG (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)); makelist(num(T(2*n-1, 1), n, 1, 5)); \\ Vladimir Kruchinin, Mar 12 2012 CROSSREFS Cf. A048604, A095885. Sequence in context: A015463 A177507 A258182 * A165210 A162454 A203210 Adjacent sequences:  A048602 A048603 A048604 * A048606 A048607 A048608 KEYWORD frac,sign,nice AUTHOR Winston C. Yang (yang(AT)math.wisc.edu) STATUS approved

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Last modified August 19 01:29 EDT 2017. Contains 290787 sequences.