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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| W. C. Yang, Composition equations, preprint, 1999.
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FORMULA
| a(x)=sum_{n=0,1,2,3...} A052136(n)/A052137(n)*x^(2n+i). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 21 2007
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MAPLE
| 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 (mathar(AT)strw.leidenuniv.nl), Jun 21 2007
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MATHEMATICA
| 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]]
(* From Jean-François Alcover, May 16 2011 *)
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CROSSREFS
| Cf. A052136. See also A048602, A048603, etc.
Sequence in context: A167893 A034723 A188685 * A003376 A156975 A081876
Adjacent sequences: A052134 A052135 A052136 * A052138 A052139 A052140
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KEYWORD
| nonn,frac,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 22 2000
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 21 2007
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