A052557 - OEIS

%I #29 Sep 08 2022 08:44:59

%S 1,0,0,6,24,120,1440,15120,161280,2177280,32659200,518918400,

%T 9101030400,174356582400,3574309939200,78460462080000,

%U 1841205510144000,45883678224384000,1210048630382592000

%N Expansion of e.g.f. (1-x)/(1-x-x^3).

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=499">Encyclopedia of Combinatorial Structures 499</a>

%F E.g.f.: (1-x)/(1 - x - x^3).

%F a(n) = n*a(n-1) + n*(n-1)*(n-2)*a(n-3), where a(0)=1, a(1)=0, a(2)=0.

%F a(n) = Sum(-1/31*(2+3*_alpha^2-11*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+_Z^3))*n!.

%F a(n) = n!*A078012(n). - _R. J. Mathar_, Nov 27 2011

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

%t With[{nn=20},CoefficientList[Series[(1-x)/(1-x-x^3),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 20 2012 *)

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( (1-x)/(1-x-x^3) )) \\ _G. C. Greubel_, May 07 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)/(1-x-x^3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 07 2019

%o (Sage)

%o R = PowerSeriesRing(QQ, 'x')

%o x = R.gen().O(30)

%o T = (1-x)/(1-x-x^3)

%o list(T.egf_to_ogf())

%o # _G. C. Greubel_, May 07 2019

%o (GAP) a:=[0,0,6];; for n in [4..30] do a[n]:=n*a[n-1]+n*(n-1)*(n-2)*a[n-3]; od; Concatenation([1], a); # _G. C. Greubel_, May 07 2019

%K easy,nonn

%O 0,4

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