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%I #8 Dec 19 2017 16:23:30
%S 1,5,109,5977,612729,100954061,24395453861,8128143367905,
%T 3571195811862385,2000535014776893973,1391684597704875555165,
%U 1177047158822263838854889,1189444022487013498606939625,1415364934488337503351305867997,1958850511524588636608881908473749
%N Expansion of e.g.f. arctanh(x*sec(x)) (odd powers only).
%F a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x*sec(x)).
%e arctanh(x*sec(x)) = x/1! + 5*x^3/3! + 109*x^5/5! + 5977*x^7/7! + 612729*x^9/9! + ...
%t nmax = 15; Table[(CoefficientList[Series[ArcTanh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%o (PARI) first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(atanh(x/cos(x)))), (4^n - 1)/3) \\ _Iain Fox_, Dec 19 2017
%Y Cf. A003700, A009118, A009119, A009562, A009563, A009765, A009843, A010050, A102075, A191003, A296465, A296467, A296741, A296742.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 19 2017
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