A003717 Expansion of e.g.f. sin(tanh(x)) (odd powers only).

1, -3, 37, -959, 41641, -2693691, 241586893, -28607094455, 4315903789009, -807258131578995, 183184249105857781, -49548882107764546223, 15742588857552887269753, -5802682207845642276301995

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

Programs

Mathematica

Sin[ Tanh[ x ] ] (* Odd Part *)
With[{nn = 60}, Take[CoefficientList[Series[Sin[Tanh[x]], {x, 0, nn}], x] Range[0, nn - 1]!, {2, -1, 2}]] (* Vincenzo Librandi, Apr 11 2014 *)

Maxima

a(n):=(-1)^(n-1)*b(2*n-1);
b(n):=sum((1-(-1)^k)/k!*((-1)^(n-k)+1)*sum(binomial(j-1,k-1)*j!*2^(n-j-2)*(-1)^((n+k)/2+j)*stirling2(n,j),j,k,n),k,1,n); /* Vladimir Kruchinin, Apr 21 2011 */

Formula

a(n) = (-1)^(n-1)*b(2*n-1), b(n) = sum(k=1..n,(1-(-1)^k)/k!*((-1)^(n-k)+1)* sum(j=k..n, binomial(j-1,k-1)*j!*2^(n-j-2)*(-1)^((n+k)/2+j)* Stirling2(n,j))). - Vladimir Kruchinin, Apr 21 2011

Extensions

Name edited by Michel Marcus, Jan 28 2018