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

1, 2, 12, 120, 1632, 27840, 570240, 13628160, 372234240, 11437977600, 390516940800, 14666390323200, 600890263142400, 26670379902566400, 1274817218759884800, 65287473566515200000, 3566486043252228096000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 628

Crossrefs

Cf. A000129, A002203, A006012.

Programs

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

Maple

spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Union(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

With[{nn=20},CoefficientList[Series[(1-2x)/(1-4x+2x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 28 2019 *)
Table[n!*2^(n/2)*ChebyshevT[n, Sqrt[2]], {n,0,50}] (* G. C. Greubel, Jun 10 2022 *)

SageMath

[2^(n/2)*factorial(n)*chebyshev_T(n, sqrt(2)) for n in (0..50)] # G. C. Greubel, Jun 10 2022

Formula

E.g.f.: (1 - 2*x)/(1 - 4*x + 2*x^2).

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).

a(n) = (n!/2)*Sum_{alpha=RootOf(1 - 4*Z + 2*Z^2)} alpha^(-n).

a(n) = n!*A006012(n). - R. J. Mathar, Nov 27 2011

From G. C. Greubel, Jun 10 2022: (Start)

a(2*n) = (2*n)! * 2^(n-1)*A002203(2*n).

a(2*n+1) = (2*n+1)! * 2^(n+1)*A000129(2*n+1).

a(n) = 2^(n/2) * n! * ChebyshevT(n, sqrt(2)). (End)