A009329 - OEIS

%I #21 Jun 27 2014 02:27:19

%S 0,1,-1,3,-10,41,-232,1299,-10064,74609,-720384,6787811,-78009600,

%T 898506649,-11977120768,163241051315,-2480763381760,39007280136801,

%U -666957211828224,11866656488375747,-225809770695098368

%N E.g.f. log(1+sin(tan(x))).

%H Vincenzo Librandi, <a href="/A009329/b009329.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 4*sum(k=0..(n-1)/2, ((-1)^(n-k+1)*sum(r=0..k, ((sum(i=0..(n-2*k)/2, (2*i-n+2*k)^(2*r+n-2*k)*(-1)^i*binomial(n-2*k,i)))*sum(j=2*r+n-2*k..n, binomial(j-1,2*r+n-2*k-1)*j!*2^(n-j-1)*(-1)^(j)*stirling2(n,j)))/(2*r+n-2*k)!))/((n-2*k)*2^(n-2*k))). - _Vladimir Kruchinin_, Jun 11 2011

%F a(n) ~ 2 * (-1)^(n+1) * (n-1)! / arctan(Pi/2)^n. - _Vaclav Kotesovec_, Jun 26 2014

%t CoefficientList[Series[Log[1+Sin[Tan[x]]], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2014 *)

%o (Maxima)

%o a(n):=4*sum(((-1)^(n-k+1)*sum(((sum((2*i-n+2*k)^(2*r+n-2*k)*(-1)^i*binomial(n-2*k,i),i,0,(n-2*k)/2))*sum(binomial(j-1,2*r+n-2*k-1)*j!*2^(n-j-1)*(-1)^(j)*stirling2(n,j),j,2*r+n-2*k,n))/(2*r+n-2*k)!,r,0,k))/((n-2*k)*2^(n-2*k)),k,0,(n-1)/2); /* _Vladimir Kruchinin_, Jun 11 2011*/

%Y Bisections are A012016 and A012240.

%K sign,easy

%O 0,4

%A _R. H. Hardin_

%E Extended with signs by _Olivier Gérard_, Mar 15 1997