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A052696 Expansion of e.g.f. (1-x)^2/(1-4*x+3*x^2-x^3).

%I A052696 #21 Jun 01 2022 01:59:49
%S A052696 1,2,12,114,1440,22680,428400,9439920,237726720,6735052800,
%T A052696 212012640000,7341338188800,277317497318400,11348577278438400,
%U A052696 500138456661043200,23615780481925632000,1189441481702842368000
%N A052696 Expansion of e.g.f. (1-x)^2/(1-4*x+3*x^2-x^3).
%H A052696 G. C. Greubel, <a href="/A052696/b052696.txt">Table of n, a(n) for n = 0..375</a>
%H A052696 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=645">Encyclopedia of Combinatorial Structures 645</a>
%F A052696 E.g.f.: (1 - x)^2/(1 - 4*x + 3*x^2 - x^3).
%F A052696 D-finite recurrence: a(0)=1, a(1)=2, a(2)=12, a(n) = 2*n*a(n-1) - 3*n*(n-1)*a(n-2) + n*(n-1)*(n-2)*a(n-3).
%F A052696 a(n) = n! * Sum_{alpha=RootOf(-1 +4*Z -3*Z^2 +Z^3)} (1/31)*(4 + 7*alpha - 2*alpha^2)*alpha^(-1-n).
%F A052696 a(n) = n! * A052544(n). - _G. C. Greubel_, May 31 2022
%p A052696 spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Z),Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052696 With[{nn=20},CoefficientList[Series[(1-x)^2/(1-4x+3x^2-x^3),{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Aug 28 2012 *)
%o A052696 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30);
%o A052696 Coefficients(R!(Laplace( (1-x)^2/(1-4*x+3*x^2-x^3) ))); // _G. C. Greubel_, May 31 2022
%o A052696 (SageMath)
%o A052696 @CachedFunction
%o A052696 def b(n): # b = A052544
%o A052696     if (n<3): return factorial(n+1)
%o A052696     else: return 4*b(n-1) - 3*b(n-2) + b(n-3)
%o A052696 def A052696(n): return factorial(n)*b(n)
%o A052696 [A052696(n) for n in (0..40)] # _G. C. Greubel_, May 31 2022
%Y A052696 Cf. A052544.
%K A052696 easy,nonn
%O A052696 0,2
%A A052696 encyclopedia(AT)pommard.inria.fr, Jan 25 2000