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%I #11 May 28 2014 15:33:03
%S 1,1,-11,-83,6921,60281,-29132611,208438245,427918448785,
%T -22588439158415,-15853957892902395,2325342085659612317,
%U 1210510298677225936025,-389238357419648883489303,-164119044571112073285613619
%N E.g.f. arctan(x/cos(x)) (odd powers only).
%F a(n)=(2*n+1)!*(2*sum(m=0..n-1, ((-1)^(m)*sum(j=0..(n-m), binomial(m+j-1/2,j)*4^(n-m-j)*sum(i=0..j, (i-j)^(2*n-2*m)*binomial(2*j,i)*(-1)^(n-m+j-i))))/((2*m+1)*(2*n+1-2*m-1)!))+(-1)^(n)/(2*n+1)).
%e arctan(x/cos(x)) = x + 1/6*x^3 - 11/120*x^5 - 83/5040*x^7 +- ...
%t With[{nn=30},Take[CoefficientList[Series[ArcTan[x/Cos[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, May 28 2014 *)
%o (Maxima)
%o a(n):=(2*n+1)!*(2*sum(((-1)^(m)*sum(binomial(m+j-1/2,j)*4^(n-m-j)*sum((i-j)^(2*n-2*m)*binomial(2*j,i)*(-1)^(n-m+j-i),i,0,j),j,0,(n-m)))/((2*m+1)*(2*n+1-2*m-1)!),m,0,n-1)+(-1)^(n)/(2*n+1));
%K sign
%O 0,3
%A _Vladimir Kruchinin_, Jun 16 2011
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