A009344 Expansion of log(1 + sin(x)^2).

0, 2, -20, 512, -27680, 2565632, -363299840, 72957489152, -19722824192000, 6905862783107072, -3040352844189532160, 1643816046204280635392, -1070739118373698870968320, 827017799903621162951770112

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Programs

Mathematica

Log[ 1+Sin[ x ]^2 ] (* Even Part *)
nn = 20; Table[(CoefficientList[Series[Log[1 + Sin[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Jan 23 2015 *)

Maxima

b(n):=sum(((-1)^(k-1)*((-1)^(n-2*k)+1)*sum((2*i-2*k)^n*binomial(2*k,i)*(-1)^((n+2*k)/2-i),i,0,k))/(k*2^(2*k)),k,1,n/2);
a(n)=b(2*n); /* Vladimir Kruchinin, Jun 28 2011 */

Formula

a(n) ~ (2*n)! * (-1)^(n+1) / (n * log(1+sqrt(2))^(2*n)). - Vaclav Kotesovec, Jan 23 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997