A052693 Expansion of e.g.f. (1-x)/(1-3*x+x^3).

1, 2, 12, 102, 1176, 16920, 292320, 5891760, 135717120, 3517032960, 101268921600, 3207514464000, 110828037196800, 4148515981209600, 167232459621427200, 7222900141416960000, 332760193091149824000

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..375
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 642

Crossrefs

Cf. A000142, A052536.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x)/(1-3*x+x^3) ))); // G. C. Greubel, Jun 01 2022

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)/(1-3x+x^3),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Dec 17 2012 *)

SageMath

@CachedFunction
def A052536(n):
    if (n<3): return factorial(n+1)
    else: return 3*A052536(n-1) - A052536(n-3)
def A052693(n): return factorial(n)*A052536(n)
[A052693(n) for n in (0..40)] # G. C. Greubel, Jun 01 2022

Formula

E.g.f.: (1-x)/(1-3*x+x^3).

Recurrence: a(0)=1, a(1)=2, a(2)=12 a(n+3) = 3*(n+3)*a(n+2) - (n+1)*(n+2)*(n+3)*a(n).

a(n) = (n!/9)*Sum_{alpha=RootOf(1 -3*Z +Z^3)} (2 - alpha + alpha^2)*alpha^(-1-n).

a(n) = n! * A052536(n). - G. C. Greubel, Jun 01 2022