|
%I #13 Jan 15 2018 15:45:14
%S 1,1,5,47,657,12245,285805,8022555,263276705,9892965545,418911700725,
%T 19738761470375,1024422336336625,58067265415960125,
%U 3569400983720767325,236508279434832201875,16804378746368557826625,1274542376742001037932625,102780751359763333970849125
%N E.g.f.: A(x) = f(x*A(x)^2), where f(x) = exp(arctan(x)).
%C 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.
%H Vaclav Kotesovec, <a href="/A088691/b088691.txt">Table of n, a(n) for n = 0..350</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of implicit functions if Fww = 0</a>
%F a(n) = n! * [x^n] (exp(arctan(x)))^(2n+1)/(2n+1).
%F 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
%t Table[n!*SeriesCoefficient[(Exp[ArcTan[x]])^(2n+1)/(2n+1),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 24 2014 *)
%o (PARI) a(n)=n!*polcoeff((exp(atan(x)))^(2*n+1)+x*O(x^n),n,x)/(2*n+1)
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 06 2003
|
