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%I #15 Sep 08 2022 08:44:38
%S 0,2,-8,272,-13568,1290752,-181296128,36501112832,-9859508338688,
%T 3453141256896512,-1520147333209653248,821912974600193441792,
%U -535368543762962928631808,413509146873290771683868672
%N Expansion of e.g.f. arctanh(sin(x)*sin(x)), even powers only.
%H G. C. Greubel, <a href="/A012301/b012301.txt">Table of n, a(n) for n = 0..220</a>
%F a(n) ~ (-1)^(n+1) * (2*n-1)! / (log(1+sqrt(2)))^(2*n). - _Vaclav Kotesovec_, Feb 08 2015
%e E.g.f. = 2*x^2/2! - 8*x^4/4! + 272*x^6/6! - 13568*x^8/8! + ...
%p seq(coeff(series(factorial(n)*arctanh(sin(x)*sin(x)),x,n+1), x, n), n = 0 .. 26,2); # _Muniru A Asiru_, Oct 26 2018
%t nn = 20; Table[(CoefficientList[Series[ArcTanh[Sin[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* _Vaclav Kotesovec_, Feb 08 2015 *)
%o (PARI) x='x+O('x^50); v=Vec(serlaplace(atanh(sin(x)*sin(x)))); concat([0], vector(#v\2,n,v[2*n-1])) \\ _G. C. Greubel_, Oct 25 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Argtanh(Sin(x)^2) )); [0] cat [Factorial(2*n+2)*b[2*n+1]: n in [0..Floor((m-4)/2)]]; // _G. C. Greubel_, Oct 25 2018
%K sign
%O 0,2
%A Patrick Demichel (patrick.demichel(AT)hp.com)
%E a(0)=0 prepended by _Vaclav Kotesovec_, Feb 08 2015
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