A003706 E.g.f. sin(tan(x)), zeros omitted.

1, 1, -3, -275, -15015, -968167, -77000363, -7433044411, -843598411471, -107426835190735, -14072980460605907, -1424712499632406371, 164163646840636339593, 237037449673450822122569, 155015924346163216960553093, 92387809011599803660871724021

Offset: 0

Author

R. H. Hardin, Simon Plouffe

Keywords

• sign

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Vladimir V. Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Programs

Mathematica

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 *)

Maxima

a(n):=b(2*n-1);
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 */
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 */

Formula

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

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

Extensions

Corrected name, Joerg Arndt, Apr 23 2011

More terms from Harvey P. Dale, Jul 25 2012