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%I #23 Jun 03 2022 19:13:27
%S 1,1,6,18,168,840,10800,75600,1249920,11249280,228614400,2514758400,
%T 60833203200,790831641600,22230464256000,333456963840000,
%U 10691545632768000,181756275757056000,6549628300959744000
%N Expansion of e.g.f. 1/((1-2*x^2)*(1-x)).
%H Vincenzo Librandi, <a href="/A052634/b052634.txt">Table of n, a(n) for n = 0..200</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=580">Encyclopedia of Combinatorial Structures 580</a>
%F E.g.f.: 1/(-1+2*x^2)/(-1+x).
%F Recurrence: {a(1)=1, a(0)=1, a(2)=6, (12+2*n^3+12*n^2+22*n)*a(n) +(-2*n^2-10*n-12)*a(n+1) +(-n-3)*a(n+2) +a(n+3)=0}.
%F a(n) = (-1+Sum(1/2*(1+2*_alpha)*_alpha^(-1-n), with _alpha=RootOf(-1+2*_Z^2)))*n! .
%F a(n) = n!*[2^floor(n/2+1)-1].
%F a(n)=n!*A052551(n). - _R. J. Mathar_, Jun 03 2022
%p spec := [S,{S=Prod(Sequence(Prod(Z,Union(Z,Z))),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t With[{nn=20},CoefficientList[Series[1/((1-2x^2)(1-x)),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 13 2014 *)
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000