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%I #18 Oct 02 2021 06:17:15
%S 0,1,2,1,-20,-151,-354,6217,100472,537777,-7631270,-223395919,
%T -2120164188,22050300505,1154262915638,17130776734905,
%U -105423782758544,-11372993234072863,-245877012220234446,345837436238423521,188329590656514108380
%N Expansion of e.g.f. arctan(x*exp(x)).
%H Seiichi Manyama, <a href="/A191719/b191719.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = n!*Sum_{m=1..(n+1)/2} ((2*m-1)^(n-2*m)*(-1)^(m-1))/(n-2*m+1)!.
%F a(n) ~ (n-1)! * sin(n*arctan(1/tan(r))) * (cos(r)/r)^n, where r = Im(LambertW(I)) = A305200 = 0.576412723031435283148289239887... is the root of the equation exp(r*tan(r))=cos(r)/r. - _Vaclav Kotesovec_, Jan 02 2014
%t Rest[CoefficientList[Series[ArcTan[x*Exp[x]],{x,0,20}],x]*Range[0,20]!] (* _Vaclav Kotesovec_, Jan 02 2014 *)
%o (Maxima)
%o a(n):=n!*sum(((2*m-1)^(n-2*m)*(-1)^(m-1))/(n-2*m+1)!,m,1,(n+1)/2);
%Y Cf. A009635, A216401, A297009.
%K sign
%O 0,3
%A _Vladimir Kruchinin_, Jun 13 2011
%E a(0)=0 prepended by _Seiichi Manyama_, Oct 01 2021
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