A052685 Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4).

1, 1, 4, 24, 168, 1680, 18720, 252000, 3830400, 65681280, 1251936000, 26225337600, 599710003200, 14851444608000, 396138155212800, 11320537003776000, 345079573622784000, 11176410365632512000

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 633

Crossrefs

Cf. A000142, A052535.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x^2)/(1-x-2*x^2+x^4) ))); // G. C. Greubel, Jun 03 2022

Maple

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

Mathematica

b[n_]:= b[n]= If[n<4, n!-2*Boole[n==3], b[n-1] +2*b[n-2] -b[n-4]]; (* b=A052535 *)
A052685[n_]:= n!*b[n];
Table[A052685[n], {n, 0, 40}] (* G. C. Greubel, Jun 03 2022 *)
With[{nn=20},CoefficientList[Series[(1-x^2)/(1-x-2 x^2+x^4),{x,0,nn}],x] Range[0,nn]!]

SageMath

@CachedFunction
def b(n): # b = A052535
    if (n<4): return factorial(n) - 2*bool(n==3)
    else: return b(n-1) + 2*b(n-2) - b(n-4)
def A052685(n): return factorial(n)*b(n)
[A052685(n) for n in (0..40)] # G. C. Greubel, Jun 03 2022

Formula

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

D-finite recurrence: a(0)=1, a(1)=1, a(2)=4, a(3)=24, a(n) = n*a(n-1) + 2*n*(n-1)*a(n-2) - n*(n-1)*(n-2)*(n-3)*a(n-4).

a(n) = (n!/283)*Sum_{alpha=RootOf(1-z-2*Z^2+Z^4)} (27 + 112*alpha + 9*alpha^2 - 48*alpha^3)*alpha^(-1-n).

a(n) = n!*A052535(n). - R. J. Mathar, Nov 27 2011