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A052137
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Denominators of power series coefficients of a(x) satisfying a(a(a(x)))= arctan(x).
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2
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1, 9, 135, 25515, 45927, 12629925, 4433103675, 1396427657625, 23739270179625, 21920842083865725, 34525326282088516875, 8734907549368394769375, 17688187787470999407984375, 413903594226821386146834375
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OFFSET
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0,2
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REFERENCES
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W. C. Yang, Composition equations, preprint, 1999.
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LINKS
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FORMULA
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MAPLE
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interface(labeling=false) : a := 0 : mPow := 15 : for i from 0 to mPow do a := a+alph[2*i+1]*x^(2*i+1) ; od: a2 := 0 : for i from 0 to mPow do a2 := a2+alph[2*i+1]*a^(2*i+1) ; od: a2 := taylor(a2, x=0, 2*mPow+2) : a2 := convert(a2, polynom) : a3 := 0 : for i from 0 to mPow do a3 := a3+alph[2*i+1]*a2^(2*i+1) ; od: for i from 0 to mPow do tanCoef[2*i+1] := coeftayl(arctan(x), x=0, 2*i+1) ; od: a3 := taylor(a3, x=0, 2*mPow+2) : a3 := convert(a3, polynom) : for i from 0 to mPow do tozer := coeftayl(a3, x=0, 2*i+1) : alph[2*i+1] := op(1, [solve(tozer=tanCoef[2*i+1], alph[2*i+1])]) : printf("%d, ", denom(alph[2*i+1])) ; ; od: # R. J. Mathar, Jun 21 2007
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MATHEMATICA
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n = 14; m = 2 n - 1 (* m = maximal degree *);
a[x_] = Sum[c[k] x^k, {k, 1, m, 2}]; coes = DeleteCases[ CoefficientList[ Series[a @ a @ a @ x - ArcTan[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] // Denominator // Partition[#, 2] &)[[All, 2]] (* Jean-François Alcover, May 16 2011 *)
T[n_, n_] = 1; T[n_, m_] := T[n, m] = 1/3*(2^(-m - 1)*m!*((-1)^(n + m) + 1)*(-1)^((3*n + m)/2)*Sum[2^i*StirlingS1[i, m]*Binomial[n - 1, i - 1]/i!, {i, m, n}] - Sum[T[k, m]*Sum[T[n, i]*T[i, k], {i, k, n}], {k, m + 1, n - 1}] - T[m, m]*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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nonn,frac,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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