A003721 Expansion of e.g.f. tan(tanh(x)) (odd powers only).

1, 0, -8, 112, -128, -109824, 8141824, -353878016, -9666461696, 5151942574080, -825073851170816, 76429076694827008, 2051308253366714368, -2361338488910424047616, 171581865952588387581952

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

• Seiichi Manyama, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A003716, A003718.

Programs

Mathematica

With[{nn=30},Take[CoefficientList[Series[Tan[Tanh[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Nov 05 2011 *)

Maxima

a(n):=sum(((sum(j!*2^(2*i+1-j-1)*(-1)^(i+j+1)*stirling2(2*i+1,j),j,1,2*i+1))*sum(binomial(k-1,2*i)*k!*(-1)^(1+k)*2^(2*n-k-1)*stirling2(2*n-1,k),k,2*i+1,2*n-1))/(2*i+1)!,i,0,(n-1)); /* Vladimir Kruchinin, Jun 10 2011 */

Formula

a(n) = Sum_{i=0..(n-1)} ( ( Sum_{j=1..2*i+1} j!*2^(2*i+1-j-1)*(-1)^(i+j+1)*Stirling2(2*i+1,j) ) * Sum_{k=2*i+1..2*n-1} binomial(k-1,2*i)*k!*(-1)^(1+k)*2^(2*n-k-1)*Stirling2(2*n-1,k) )/(2*i+1)!. - Vladimir Kruchinin, Jun 10 2011