A052677 Expansion of e.g.f. (1-x)/(1-4*x+x^2).

1, 3, 22, 246, 3672, 68520, 1534320, 40083120, 1196737920, 40196580480, 1500156806400, 61585275628800, 2758072531737600, 133812468652262400, 6991529043750451200, 391391124208051968000, 23371064978815217664000

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..365
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 625

Crossrefs

Cf. A000142, A079935.

Programs

Magma

[n le 2 select 3^(n-1) else 4*(n-1)*Self(n-1) - (n-1)*(n-2)*Self(n-2): n in [1..31]]; // G. C. Greubel, Jun 11 2022

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)/(1-4x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 28 2016 *)

SageMath

[factorial(n)*sum( binomial(2*n-k, k)*2^(n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 11 2022

Formula

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

Recurrence: a(0)=1, a(1)=3, a(n+2) = 4*(n+2)*a(n+1) - (n+2)*(n+1)*a(n).

a(n) = (n!/6)*Sum_{alpha=RootOf(1 -4*Z +Z^2)} (1 + alpha)*alpha^(-1-n).

a(n) = n!*A079935(n). - R. J. Mathar, Nov 27 2011

a(n) = (-1)^n * n! * ChebyshevU(2*n, i/sqrt(2)). - G. C. Greubel, Jun 11 2022