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

%I #25 Mar 09 2023 15:40:49

%S 1,2,10,60,504,5040,61200,856800,13749120,247484160,4953312000,

%T 108972864000,2615827737600,68011521177600,1904409771264000,

%U 57132293137920000,1828254303203328000,62160646308913152000

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

%H G. C. Greubel, <a href="/A052600/b052600.txt">Table of n, a(n) for n = 0..400</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=545">Encyclopedia of Combinatorial Structures 545</a>

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

%F Recurrence: {a(0)=1, a(1)=2, a(2)=10, (12+2*n^3+12*n^2+22*n)*a(n) +(-n^2-5*n-6)*a(n+1) +(-2*n-6)*a(n+2) +a(n+3)=0}.

%F a(n) = (4/3*2^n+Sum_(-1/6*(2+_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2)))*n!

%F a(n) = (4*2^n-1)/3*n! if n is even, a(n) = (4*2^n-2)/3*n! otherwise.

%F a(n) = n!*A000975(n+1). - _R. J. Mathar_

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

%t With[{nn=20},CoefficientList[Series[1/((1-2x)(1-x^2)),{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Jan 21 2013 *)

%o (PARI) x='x+O('x^50); Vec(serlaplace(1/((1-2*x)*(1-x^2)))) \\ _G. C. Greubel_, Oct 11 2017

%Y Cf. A000975.

%K easy,nonn

%O 0,2

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