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A052616 Expansion of e.g.f. (3+2*x)/(1-x^2).

%I A052616 #25 Feb 02 2023 05:04:18
%S A052616 3,2,6,12,72,240,2160,10080,120960,725760,10886400,79833600,
%T A052616 1437004800,12454041600,261534873600,2615348736000,62768369664000,
%U A052616 711374856192000,19207121117184000,243290200817664000,7298706024529920000,102181884343418880000,3372002183332823040000
%N A052616 Expansion of e.g.f. (3+2*x)/(1-x^2).
%H A052616 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=561">Encyclopedia of Combinatorial Structures 561</a>.
%F A052616 E.g.f.: (2*x+3)/(1-x^2).
%F A052616 Recurrence: {a(1)=2, a(0)=3, (-2-n^2-3*n)*a(n) + a(n+2) = 0}.
%F A052616 a(n) = Sum(1/2*(3*_alpha+2)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))*n!.
%F A052616 a(n) = 3n! if n is even, 2n! otherwise.
%F A052616 a(n) = n!*A176059(n). - _R. J. Mathar_, Jun 03 2022
%F A052616 Sum_{n>=0} 1/a(n) = (5*e^2-1)/(12*e) = cosh(1)/3 + sinh(1)/2. - _Amiram Eldar_, Feb 02 2023
%p A052616 spec := [S,{S=Union(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052616 With[{nn=30},CoefficientList[Series[(3+2x)/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 12 2021 *)
%Y A052616 Cf. A176059.
%K A052616 easy,nonn
%O A052616 0,1
%A A052616 encyclopedia(AT)pommard.inria.fr, Jan 25 2000