A003716 Expansion of e.g.f. tan(sinh(x)) (odd powers only).

1, 3, 37, 1015, 47881, 3459819, 354711853, 48961863007, 8754050024209, 1967989239505875, 543326939019354421, 180718022989699819207, 71275877445849484090393, 32890432371345908634652347, 17555593768891213894861569085

Offset: 0

Author

R. H. Hardin

Keywords

• nonn

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. A003718, A003721.

Programs

Mathematica

Tan[ Sinh[ x ] ] (* Odd Part *)
nn = 20; Table[(CoefficientList[Series[Tan[Sinh[x]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)

Maxima

a(n):=sum(((-1)^(k-1)+1)/(2^k*k!)*sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)*(sum(j!*2^(k-j-1)*(-1)^((k+1)/2+j)*stirling2(k,j),j,1,k)),k,1,n); /* Vladimir Kruchinin, Apr 20 2011 */

Formula

a(n) = Sum_{k=1..n} ((-1)^(k-1)+1)/(2^k*k!) * ( Sum_{i=0..k} (-1)^i*(k-2*i)^n *binomial(k,i) ) * ( Sum_{j=1..k} j! * 2^(k-j-1) * (-1)^((k+1)/2+j) * stirling2(k,j) ). - Vladimir Kruchinin, Apr 20 2011

a(n) ~ 4 * (2*n+1)! / (sqrt(4+Pi^2) * (log((Pi + sqrt(4+Pi^2))/2))^(2*n+2)). - Vaclav Kotesovec, Feb 16 2015