A009273 Expansion of e.g.f. exp(x*tanh(x)) (even powers only).

1, 2, 4, -24, 400, -5600, -103872, 26975872, -3438685952, 417995260416, -51382607559680, 5994623640856576, -454930757753597952, -94991612229069430784, 81515752167646959124480, -41079088828539119883878400, 18870487103065970636941754368, -8553231336572387307575081566208

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..240

Crossrefs

Cf. A354245, A354399.

Programs

Mathematica

nmax = 20; Table[(CoefficientList[Series[E^(x*Tanh[x]), {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}] (* Vaclav Kotesovec, May 24 2022 *)
With[{nn=40},Take[CoefficientList[Series[Exp[x Tanh[x]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Apr 02 2025 *)

Maxima

a(n):=sum(binomial(2*n,m)*sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(2*n-2*m-k)*stirling2(2*n-m,k+m),k,0,2*n-2*m),m,0,2*n); /* Vladimir Kruchinin, Jun 06 2011 */

Formula

a(n) = Sum_(m=0..2*n, binomial(2*n,m)*Sum_(k=0..2*n-2*m, binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(2*n-2*m-k)*stirling2(2*n-m,k+m))), n>0, a(0)=1. - Vladimir Kruchinin, Jun 06 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

More terms from Vaclav Kotesovec, May 25 2022