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Expansion of e.g.f. (2-x-2x^2)/((1-x)(1-2x^2)).
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%I #18 Sep 07 2022 22:24:25

%S 2,1,6,6,120,120,6480,5040,685440,362880,119750400,39916800,

%T 31135104000,6227020800,11245999564800,1307674368000,5377157001216000,

%U 355687428096000,3284417711038464000

%N Expansion of e.g.f. (2-x-2x^2)/((1-x)(1-2x^2)).

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=582">Encyclopedia of Combinatorial Structures 582</a>

%F E.g.f.: -(-2+x+2*x^2)/(-1+2*x^2)/(-1+x).

%F D-finite Recurrence: {a(1)=1, a(2)=6, a(0)=2, (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 (1+Sum(1/2*_alpha^(-n), _alpha=RootOf(-1+2*_Z^2)))*n!

%F n!*[2^(n/2)+1] if n is even, n! otherwise.

%F a(n) = n!*A052552(n). - _R. J. Mathar_, Jun 03 2022

%p spec := [S,{S=Union(Sequence(Prod(Z,Union(Z,Z))), Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t With[{nn=20},CoefficientList[Series[(2-x-2x^2)/((1-x)(1-2x^2)),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 16 2016 *)

%Y Cf. A052552.

%K easy,nonn

%O 0,1

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000