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A052684 Expansion of e.g.f. 1/(1-2*x^2-x^3).

%I A052684 #25 Jun 01 2025 18:39:01
%S A052684 1,0,4,6,96,480,6480,60480,887040,11975040,203212800,3512678400,
%T A052684 69455232000,1444668825600,32953394073600,796373690112000,
%U A052684 20671716409344000,567677135241216000,16550136029306880000
%N A052684 Expansion of e.g.f. 1/(1-2*x^2-x^3).
%H A052684 G. C. Greubel, <a href="/A052684/b052684.txt">Table of n, a(n) for n = 0..350</a>
%H A052684 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=632">Encyclopedia of Combinatorial Structures 632</a>
%F A052684 E.g.f.: 1/(1 - 2*x^2 - x^3).
%F A052684 D-finite recurrence: a(0)=1, a(1)=0, a(2)=4, a(n) = 2*n*(n-1)*a(n-2) + n*(n-1)*(n-2)*a(n-3).
%F A052684 a(n) = (n!/5)*Sum_{alpha=RootOf(-1+2*Z^2+Z^3)} (-6 + 7*alpha + 8*alpha^2)*alpha^(-1-n).
%F A052684 a(n) = n!*A008346(n). - _R. J. Mathar_, Nov 27 2011
%p A052684 spec := [S,{S=Sequence(Prod(Z,Union(Z,Z,Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052684 With[{nn=20},CoefficientList[Series[1/(1-2x^2-x^3),{x,0,nn}], x] Range[ 0,nn]!] (* _Harvey P. Dale_, Apr 22 2012 *)
%t A052684 Table[n!*(Fibonacci[n]+(-1)^n), {n,0,40}] (* _G. C. Greubel_, Jun 03 2022 *)
%o A052684 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 1/(1-2*x^2-x^3) ))); // _G. C. Greubel_, Jun 03 2022
%o A052684 (SageMath) [factorial(n)*(fibonacci(n) +(-1)^n) for n in (0..40)] # _G. C. Greubel_, Jun 03 2022
%Y A052684 Cf. A000142, A008346.
%K A052684 easy,nonn
%O A052684 0,3
%A A052684 encyclopedia(AT)pommard.inria.fr, Jan 25 2000