A191003 E.g.f. arctan(x/cos(x)) (odd powers only).

1, 1, -11, -83, 6921, 60281, -29132611, 208438245, 427918448785, -22588439158415, -15853957892902395, 2325342085659612317, 1210510298677225936025, -389238357419648883489303, -164119044571112073285613619

Offset: 0

Author

Vladimir Kruchinin, Jun 16 2011

Keywords

• sign

Examples

			arctan(x/cos(x)) = x + 1/6*x^3 - 11/120*x^5 - 83/5040*x^7 +- ...
		

Programs

Mathematica

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 *)

Maxima

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));

Formula

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)).