A052684 Expansion of e.g.f. 1/(1-2*x^2-x^3).

1, 0, 4, 6, 96, 480, 6480, 60480, 887040, 11975040, 203212800, 3512678400, 69455232000, 1444668825600, 32953394073600, 796373690112000, 20671716409344000, 567677135241216000, 16550136029306880000

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 632

Crossrefs

Cf. A000142, A008346.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 1/(1-2*x^2-x^3) ))); // G. C. Greubel, Jun 03 2022

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[1/(1-2x^2-x^3),{x,0,nn}], x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 22 2012 *)
Table[n!*(Fibonacci[n]+(-1)^n), {n,0,40}] (* G. C. Greubel, Jun 03 2022 *)

SageMath

[factorial(n)*(fibonacci(n) +(-1)^n) for n in (0..40)] # G. C. Greubel, Jun 03 2022

Formula

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

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

a(n) = (n!/5)*Sum_{alpha=RootOf(-1+2*Z^2+Z^3)} (-6 + 7*alpha + 8*alpha^2)*alpha^(-1-n).

a(n) = n!*A008346(n). - R. J. Mathar, Nov 27 2011