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A052685 Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).

%I A052685 #21 May 21 2023 15:29:30
%S A052685 1,1,4,24,168,1680,18720,252000,3830400,65681280,1251936000,
%T A052685 26225337600,599710003200,14851444608000,396138155212800,
%U A052685 11320537003776000,345079573622784000,11176410365632512000
%N A052685 Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).
%H A052685 G. C. Greubel, <a href="/A052685/b052685.txt">Table of n, a(n) for n = 0..350</a>
%H A052685 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=633">Encyclopedia of Combinatorial Structures 633</a>
%F A052685 E.g.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
%F A052685 D-finite recurrence: a(0)=1, a(1)=1, a(2)=4, a(3)=24, a(n) = n*a(n-1) + 2*n*(n-1)*a(n-2) - n*(n-1)*(n-2)*(n-3)*a(n-4).
%F A052685 a(n) = (n!/283)*Sum_{alpha=RootOf(1-z-2*Z^2+Z^4)} (27 + 112*alpha + 9*alpha^2 - 48*alpha^3)*alpha^(-1-n).
%F A052685 a(n) = n!*A052535(n). - _R. J. Mathar_, Nov 27 2011
%p A052685 spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Prod(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052685 b[n_]:= b[n]= If[n<4, n!-2*Boole[n==3], b[n-1] +2*b[n-2] -b[n-4]]; (* b=A052535 *)
%t A052685 A052685[n_]:= n!*b[n];
%t A052685 Table[A052685[n], {n, 0, 40}] (* _G. C. Greubel_, Jun 03 2022 *)
%t A052685 With[{nn=20},CoefficientList[Series[(1-x^2)/(1-x-2 x^2+x^4),{x,0,nn}],x] Range[0,nn]!]
%o A052685 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x^2)/(1-x-2*x^2+x^4) ))); // _G. C. Greubel_, Jun 03 2022
%o A052685 (SageMath)
%o A052685 @CachedFunction
%o A052685 def b(n): # b = A052535
%o A052685     if (n<4): return factorial(n) - 2*bool(n==3)
%o A052685     else: return b(n-1) + 2*b(n-2) - b(n-4)
%o A052685 def A052685(n): return factorial(n)*b(n)
%o A052685 [A052685(n) for n in (0..40)] # _G. C. Greubel_, Jun 03 2022
%Y A052685 Cf. A000142, A052535.
%K A052685 easy,nonn
%O A052685 0,3
%A A052685 encyclopedia(AT)pommard.inria.fr, Jan 25 2000