A052658 Expansion of e.g.f. (1-x^2)*(1-x)/(1-2x-x^2+x^3).

1, 1, 4, 30, 264, 3000, 40320, 635040, 11410560, 230791680, 5185555200, 128172844800, 3455996544000, 100952461209600, 3175730791833600, 107037070043904000, 3848161361780736000, 146994587721805824000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 605

Programs

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[((1-x^2)(1-x))/(1-2x-x^2+x^3),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, May 16 2012 *)

Formula

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

Recurrence: {a(1)=1, a(0)=1, a(2)=4, (n^3+6*n^2+11*n+6)*a(n)+(-n^2-5*n-6)*a(n+1)+(-2*n-6)*a(n+2)+a(n+3)=0, a(3)=30}

a(n) = Sum(-1/7*(_alpha+_alpha^2-2)*_alpha^(-1-n), _alpha=RootOf(_Z^3-_Z^2-2*_Z+1))*n!.

a(n) = n!*A006054(n+1),n>0. - R. J. Mathar, Jun 03 2022