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A088691 E.g.f.: A(x) = f(x*A(x)^2), where f(x) = exp(arctan(x)). 1
1, 1, 5, 47, 657, 12245, 285805, 8022555, 263276705, 9892965545, 418911700725, 19738761470375, 1024422336336625, 58067265415960125, 3569400983720767325, 236508279434832201875, 16804378746368557826625, 1274542376742001037932625, 102780751359763333970849125 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
FORMULA
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
MATHEMATICA
Table[n!*SeriesCoefficient[(Exp[ArcTan[x]])^(2n+1)/(2n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)
PROG
(PARI) a(n)=n!*polcoeff((exp(atan(x)))^(2*n+1)+x*O(x^n), n, x)/(2*n+1)
CROSSREFS
Sequence in context: A367079 A180254 A127696 * A052802 A098799 A270529
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 06 2003
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)