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A052693 Expansion of e.g.f. (1-x)/(1-3*x+x^3).

%I A052693 #19 Jun 08 2022 03:27:36
%S A052693 1,2,12,102,1176,16920,292320,5891760,135717120,3517032960,
%T A052693 101268921600,3207514464000,110828037196800,4148515981209600,
%U A052693 167232459621427200,7222900141416960000,332760193091149824000
%N A052693 Expansion of e.g.f. (1-x)/(1-3*x+x^3).
%H A052693 G. C. Greubel, <a href="/A052693/b052693.txt">Table of n, a(n) for n = 0..375</a>
%H A052693 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=642">Encyclopedia of Combinatorial Structures 642</a>
%F A052693 E.g.f.: (1-x)/(1-3*x+x^3).
%F A052693 Recurrence: a(0)=1, a(1)=2, a(2)=12 a(n+3) = 3*(n+3)*a(n+2) - (n+1)*(n+2)*(n+3)*a(n).
%F A052693 a(n) = (n!/9)*Sum_{alpha=RootOf(1 -3*Z +Z^3)} (2 - alpha + alpha^2)*alpha^(-1-n).
%F A052693 a(n) = n! * A052536(n). - _G. C. Greubel_, Jun 01 2022
%p A052693 spec := [S,{S=Sequence(Union(Z,Prod(Z,Union(Z,Sequence(Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052693 With[{nn=20},CoefficientList[Series[(1-x)/(1-3x+x^3),{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Dec 17 2012 *)
%o A052693 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x)/(1-3*x+x^3) ))); // _G. C. Greubel_, Jun 01 2022
%o A052693 (SageMath)
%o A052693 @CachedFunction
%o A052693 def A052536(n):
%o A052693     if (n<3): return factorial(n+1)
%o A052693     else: return 3*A052536(n-1) - A052536(n-3)
%o A052693 def A052693(n): return factorial(n)*A052536(n)
%o A052693 [A052693(n) for n in (0..40)] # _G. C. Greubel_, Jun 01 2022
%Y A052693 Cf. A000142, A052536.
%K A052693 easy,nonn
%O A052693 0,2
%A A052693 encyclopedia(AT)pommard.inria.fr, Jan 25 2000