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A088691
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E.g.f.: A(x) = f(x*A(x)^2), where f(x) = exp(arctan(x)).
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1
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1, 1, 5, 47, 657, 12245, 285805, 8022555, 263276705, 9892965545, 418911700725, 19738761470375, 1024422336336625, 58067265415960125, 3569400983720767325, 236508279434832201875, 16804378746368557826625, 1274542376742001037932625, 102780751359763333970849125
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OFFSET
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0,3
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COMMENTS
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Radius of convergence of A(x): r = exp(-Pi/2) = 0.207879576..., with A(r) = exp(Pi/4) = 2.19328..., where r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula based on an heuristic argument.
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LINKS
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FORMULA
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a(n) = n! * [x^n] (exp(arctan(x)))^(2n+1)/(2n+1).
a(n) ~ GAMMA(1/3) * exp(n*(Pi/2-1) + Pi/4) * n^(n-5/6) / (2*6^(1/6)*sqrt(Pi)) * (1 - c/n^(1/3)), where c = 0.4593... - Vaclav Kotesovec, Jan 24 2014
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MATHEMATICA
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Table[n!*SeriesCoefficient[(Exp[ArcTan[x]])^(2n+1)/(2n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)
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PROG
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(PARI) a(n)=n!*polcoeff((exp(atan(x)))^(2*n+1)+x*O(x^n), n, x)/(2*n+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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