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%I M3176 #37 Feb 08 2015 08:47:12
%S 1,1,-3,-275,-15015,-968167,-77000363,-7433044411,-843598411471,
%T -107426835190735,-14072980460605907,-1424712499632406371,
%U 164163646840636339593,237037449673450822122569,155015924346163216960553093,92387809011599803660871724021
%N E.g.f. sin(tan(x)), zeros omitted.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A003706/b003706.txt">Table of n, a(n) for n = 0..100</a>
%H Vladimir V. Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F a(n) = b(2*n-1), b(n) = sum(k=1..n, ((-1)^(n-k)+1)*sum(j=k..n, binomial(j-1,k-1)*j!*2^(n-j-1)*(-1)^((n+k)/2+j)*stirling2(n,j))*((-1)^((k+3)/2)-(-1)^((3*k+3)/2))/(2*k!)). - _Vladimir Kruchinin_, Apr 23 2011
%F a(n) = sum(m=0..n, sum(j=0..2*n-2*m, binomial(j+2*m,2*m)*(j+2*m+1)!*2^(2*n-2*m-j)*(-1)^(n+j)* stirling2(2*n+1,j+2*m+1))/((2*m+1)!)). - _Vladimir Kruchinin_, Jan 21 2012
%t With[{nn=50},Take[CoefficientList[Series[Sin[Tan[x]],{x,0,nn}], x] Range[ 0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Jul 25 2012 *)
%o (Maxima)
%o a(n):=b(2*n-1);
%o b(n):=sum(((-1)^(n-k)+1)*sum(binomial(j-1,k-1)*j!*2^(n-j-1)*(-1)^((n+k)/2+j)*stirling2(n,j),j,k,n)*((-1)^((k+3)/2)-(-1)^((3*k+3)/2))/(2*k!),k,1,n); /* _Vladimir Kruchinin_, Apr 23 2011 */
%o a(n):=sum(sum(binomial(j+2*m,2*m)*(j+2*m+1)!*2^(2*n-2*m-j)*(-1)^(n+j)*stirling2(2*n+1,j+2*m+1),j,0,2*n-2*m)/((2*m+1)!),m,0,n); /* _Vladimir Kruchinin_, Jan 21 2012 */
%K sign
%O 0,3
%A _R. H. Hardin_, _Simon Plouffe_
%E Corrected name, _Joerg Arndt_, Apr 23 2011
%E More terms from _Harvey P. Dale_, Jul 25 2012
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