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%I #20 Sep 08 2022 08:44:38
%S 0,1,2,0,-24,-140,-8,8960,92736,12240,-14154208,-210017280,-50776704,
%T 58549241920,1133642065792,435228385280,-500888609593344,
%U -11981516394489600,-6690495105515008,7684815276420464640
%N Expansion of e.g.f. arctan(sin(x)*exp(x)).
%H G. C. Greubel, <a href="/A012290/b012290.txt">Table of n, a(n) for n = 0..423</a>
%F a(n) = Sum_{m=0..n/2} ((4^(-m)*Sum_{j=m..n/2} (binomial(n+1,2*j+1)*((2*m+1)^(n-2*j-1)*sum(i=0..m+1/2, (2*i-2*m-1)^(2*j+1)*(-1)^(j+1-i)*binomial(2*m+1,i)))))). - _Vladimir Kruchinin_, Jun 30 2011
%e arctan(sin(x)*exp(x)) = x + (2/2!)*x^2 - (24/4!)*x^4 - (140/5!)*x^5 - (8/6!)*x^6 + ...
%t CoefficientList[Series[ArcTan[Sin[x]*Exp[x]],{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Jan 02 2014 *)
%o (Maxima)
%o a(n):=sum((4^(-m)*sum(binomial(n+1,2*j+1)*((2*m+1)^(n-2*j-1)*sum((2*i-2*m-1)^(2*j+1)*(-1)^(j+1-i)*binomial(2*m+1,i),i,0,m+1/2)),j,m,(n)/2)),m,0,(n)/2); /* _Vladimir Kruchinin_, Jun 30 2011 */
%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(atan(sin(x)*exp(x))))) \\ _G. C. Greubel_, Oct 26 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Arctan(Sin(x)*Exp(x)) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // _G. C. Greubel_, Oct 26 2018
%K sign
%O 0,3
%A Patrick Demichel (patrick.demichel(AT)hp.com)
%E Missing a(0)=0 prepended by _Vaclav Kotesovec_, Jan 02 2014
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