A052674 Expansion of e.g.f. (1-x)/(1-3*x-2*x^2+2*x^3).

1, 2, 16, 156, 2160, 36720, 753120, 17992800, 491500800, 15102339840, 515630707200, 19365156518400, 793401964185600, 35214960849868800, 1683239666985676800, 86204093846846976000, 4709107007890661376000, 273324248772505362432000, 16797372435596048744448000

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 622

Crossrefs

Cf. A000142, A052543.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x)/((1+x)*(1-4*x+2*x^2)) ))); // G. C. Greubel, Jun 12 2022

Maple

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

Mathematica

Table[(n!/7)*(2*(-1)^n + 2^(n/2)*(5*ChebyshevU[n, Sqrt[2]] - 2*Sqrt[2]*ChebyshevU[n - 1, Sqrt[2]])), {n, 0, 30}] (* G. C. Greubel, Jun 12 2022 *)

SageMath

[factorial(n)*(2^(n/2)*(5*chebyshev_U(n,sqrt(2)) - 2*sqrt(2)*chebyshev_U(n-1, sqrt(2))) + 2*(-1)^n)/7 for n in (0..30)] # G. C. Greubel, Jun 12 2022

Formula

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

Recurrence: a(0)=1, a(1)=2, a(2)=16, a(n) = 3*n*a(n-1) + 2*n*(n-1)*a(n-2) - 2*n*(n-1)*(n-2)*a(n-3).

a(n) = (n!/98)*Sum_{alpha=RootOf(1 -3*Z -2*Z^2 +2*Z^3)} (13 + 25*alpha - 16*alpha^2)*alpha^(-1-n).

a(n) = n!*A052543(n). - R. J. Mathar, Nov 27 2011

a(n) = (n!/7)*(2*(-1)^n + 2^(n/2)*( 5*ChebyshevU(n, sqrt(2)) - 2*sqrt(2)*ChebyshevU(n-1, sqrt(2)) )). - G. C. Greubel, Jun 12 2022