A052696 Expansion of e.g.f. (1-x)^2/(1-4*x+3*x^2-x^3).

1, 2, 12, 114, 1440, 22680, 428400, 9439920, 237726720, 6735052800, 212012640000, 7341338188800, 277317497318400, 11348577278438400, 500138456661043200, 23615780481925632000, 1189441481702842368000

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 645

Crossrefs

Cf. A052544.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( (1-x)^2/(1-4*x+3*x^2-x^3) ))); // G. C. Greubel, May 31 2022

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)^2/(1-4x+3x^2-x^3),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Aug 28 2012 *)

SageMath

@CachedFunction
def b(n): # b = A052544
    if (n<3): return factorial(n+1)
    else: return 4*b(n-1) - 3*b(n-2) + b(n-3)
def A052696(n): return factorial(n)*b(n)
[A052696(n) for n in (0..40)] # G. C. Greubel, May 31 2022

Formula

E.g.f.: (1 - x)^2/(1 - 4*x + 3*x^2 - x^3).

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

a(n) = n! * Sum_{alpha=RootOf(-1 +4*Z -3*Z^2 +Z^3)} (1/31)*(4 + 7*alpha - 2*alpha^2)*alpha^(-1-n).

a(n) = n! * A052544(n). - G. C. Greubel, May 31 2022