OFFSET
0,3
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)).
EXAMPLE
arctan(x/cos(x)) = x + 1/6*x^3 - 11/120*x^5 - 83/5040*x^7 +- ...
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 *)
PROG
(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));
CROSSREFS
KEYWORD
sign
AUTHOR
Vladimir Kruchinin, Jun 16 2011
STATUS
approved