A009273 - OEIS

%I #28 May 25 2022 09:26:04

%S 1,2,4,-24,400,-5600,-103872,26975872,-3438685952,417995260416,

%T -51382607559680,5994623640856576,-454930757753597952,

%U -94991612229069430784,81515752167646959124480,-41079088828539119883878400,18870487103065970636941754368,-8553231336572387307575081566208

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

%H Vaclav Kotesovec, <a href="/A009273/b009273.txt">Table of n, a(n) for n = 0..240</a>

%F 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

%t 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 *)

%o (Maxima)

%o 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 */

%Y Cf. A354245, A354399.

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier Gérard_, Mar 15 1997

%E More terms from _Vaclav Kotesovec_, May 25 2022