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%I #9 Dec 19 2017 16:22:44
%S 1,4,64,2752,237312,34390016,7512117248,2302977392640,942529341030400,
%T 496287845973753856,326775812392982937600,263039306566659448242176,
%U 254121613033387345942937600,290175686081926976733941071872,386599796043915196967089006968832
%N Expansion of e.g.f. arcsin(x*sec(x)) (odd powers only).
%F a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x*sec(x)).
%e arcsin(x*sec(x)) = x/1! + 4*x^3/3! + 64*x^5/5! + 2752*x^7/7! + 237312*x^9/9! + ...
%t nmax = 15; Table[(CoefficientList[Series[ArcSin[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(asin(x/cos(x)))), (4^n - 1)/3) \\ _Iain Fox_, Dec 19 2017
%Y Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296742, A296743.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 19 2017
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