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A052680 Expansion of e.g.f. (1-2*x)/(1-4*x+2*x^2).

%I A052680 #21 Jun 10 2022 06:16:07
%S A052680 1,2,12,120,1632,27840,570240,13628160,372234240,11437977600,
%T A052680 390516940800,14666390323200,600890263142400,26670379902566400,
%U A052680 1274817218759884800,65287473566515200000,3566486043252228096000
%N A052680 Expansion of e.g.f. (1-2*x)/(1-4*x+2*x^2).
%H A052680 G. C. Greubel, <a href="/A052680/b052680.txt">Table of n, a(n) for n = 0..350</a>
%H A052680 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=628">Encyclopedia of Combinatorial Structures 628</a>
%F A052680 E.g.f.: (1 - 2*x)/(1 - 4*x + 2*x^2).
%F A052680 D-finite with Recurrence: a(0)=1, a(1)=2, a(n+2) = 4*(n+2)*a(n+1) - 2*(2 +3*n +n^2)*a(n).
%F A052680 a(n) = (n!/2)*Sum_{alpha=RootOf(1 - 4*Z + 2*Z^2)} alpha^(-n).
%F A052680 a(n) = n!*A006012(n). - _R. J. Mathar_, Nov 27 2011
%F A052680 From _G. C. Greubel_, Jun 10 2022: (Start)
%F A052680 a(2*n) = (2*n)! * 2^(n-1)*A002203(2*n).
%F A052680 a(2*n+1) = (2*n+1)! * 2^(n+1)*A000129(2*n+1).
%F A052680 a(n) = 2^(n/2) * n! * ChebyshevT(n, sqrt(2)). (End)
%p A052680 spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Union(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052680 With[{nn=20},CoefficientList[Series[(1-2x)/(1-4x+2x^2),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jan 28 2019 *)
%t A052680 Table[n!*2^(n/2)*ChebyshevT[n, Sqrt[2]], {n,0,50}] (* _G. C. Greubel_, Jun 10 2022 *)
%o A052680 (Magma) [Factorial(n)*(&+[Binomial(n, 2*k)*2^(n-k): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jun 10 2022
%o A052680 (SageMath) [2^(n/2)*factorial(n)*chebyshev_T(n, sqrt(2)) for n in (0..50)] # _G. C. Greubel_, Jun 10 2022
%Y A052680 Cf. A000129, A002203, A006012.
%K A052680 easy,nonn
%O A052680 0,2
%A A052680 encyclopedia(AT)pommard.inria.fr, Jan 25 2000