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%I #15 Jan 30 2020 21:29:16
%S 1,-1,0,6,-24,0,720,-5040,0,362880,-3628800,0,479001600,-6227020800,0,
%T 1307674368000,-20922789888000,0,6402373705728000,-121645100408832000,
%U 0,51090942171709440000,-1124000727777607680000
%N E.g.f. 2*sqrt(3)/3*arctan(sqrt(3)*x/(x+2)).
%F a(3*n+1) = (3*n)!, a(3*n+2) = -(3*n+1)!, a(3*n) = 0.
%F E.g.f.: A(x) = 2*sqrt(3)/3*arctan(sqrt(3)*x/(x+2)) = x-x^2/2!+6*x^4/4!-24*x^5/5!+720*x^7/7!-....
%F The derivative A'(x) = 1/(1+x+x^2). The inverse function A^-1(x) = 2/sqrt(3)*tan(sqrt(3)/2*x)/(1-1/sqrt(3)*tan(sqrt(3)/2*x)) is the generating function for A080635 (apart from the initial term).
%F D-finite with recurrence: a(n) +(n-1)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - _R. J. Mathar_, Jan 25 2020
%t With[{nn=30},Rest[CoefficientList[Series[2 Sqrt[3]/3 ArcTan[Sqrt[ 3] x/(x+2)],{x,0,nn}],x] Range[0,nn-1]!]] (* _Harvey P. Dale_, May 13 2019 *)
%Y A080635
%K sign,easy
%O 1,4
%A _Peter Bala_, Sep 02 2011
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