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

%I #19 Apr 01 2017 14:05:00

%S 0,2,-20,512,-27680,2565632,-363299840,72957489152,-19722824192000,

%T 6905862783107072,-3040352844189532160,1643816046204280635392,

%U -1070739118373698870968320,827017799903621162951770112

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

%F a(n) ~ (2*n)! * (-1)^(n+1) / (n * log(1+sqrt(2))^(2*n)). - _Vaclav Kotesovec_, Jan 23 2015

%t Log[ 1+Sin[ x ]^2 ] (* Even Part *)

%t 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 *)

%o (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);

%o a(n)=b(2*n); /* _Vladimir Kruchinin_, Jun 28 2011 */

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997