A052559 Expansion of e.g.f. (1-x)/(1 - 2*x - x^2 + x^3).

1, 1, 6, 36, 336, 3720, 50400, 791280, 14232960, 287763840, 6466521600, 159826867200, 4309577395200, 125885452492800, 3960073877760000, 133473015067392000, 4798579092443136000, 183299247820136448000

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..395
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 501

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)/(1-2*x-x^2+x^3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 06 2019

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x-x^2+x^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2018 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace( (1-x)/(1-2*x-x^2+x^3) )) \\ G. C. Greubel, May 06 2019

Sage

m = 30; T = taylor((1-x)/(1-2*x-x^2+x^3), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 06 2019

Formula

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

a(n) = 2*n*a(n-1) + n*(n-1)*a(n-2) - n*(n-1)*(n-2)*a(n-3), with a(0)=1, a(1)=1, a(2)=6.

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

a(n) = n!*A077998(n). - R. J. Mathar, Nov 27 2011