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%I #16 Sep 08 2022 08:44:38
%S 0,1,2,4,24,180,1432,14544,176064,2382800,36330272,618520384,
%T 11562021504,235623136320,5205288291712,123834383495424,
%U 3155999144761344,85799392788650240,2478387574846218752
%N Expansion of e.g.f. arctanh(sin(x)*exp(x)) = x+2/2!*x^2+4/3!*x^3+24/4!*x^4+180/5!*x^5...
%H G. C. Greubel, <a href="/A012292/b012292.txt">Table of n, a(n) for n = 0..414</a>
%F a(n) ~ 1/2 * (n-1)! / r^n, where r = 0.588532743981861... is the real root of the equation sin(r) = exp(-r). - _Vaclav Kotesovec_, Oct 25 2013
%e E.g.f. = x + 2*x^2/2! + 4*x^3/3! + 24*x^4/4! + 180*x^5/5! + ...
%t CoefficientList[Series[ArcTanh[Sin[x]*Exp[x]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 24 2013 *)
%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(atanh(sin(x)*exp(x))))) \\ _G. C. Greubel_, Oct 26 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Argtanh(Sin(x)*Exp(x)) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // _G. C. Greubel_, Oct 26 2018
%K nonn
%O 0,3
%A Patrick Demichel (patrick.demichel(AT)hp.com)
%E Prepended missing a(0)=0, _Vaclav Kotesovec_, Oct 24 2013
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