A052691 - OEIS

%I #21 Jun 08 2022 03:27:19

%S 1,1,2,12,96,840,8640,105840,1491840,23587200,413683200,7983360000,

%T 168129561600,3835844812800,94239732787200,2480658276096000,

%U 69651967537152000,2077925954936832000,65637135231123456000

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

%H G. C. Greubel, <a href="/A052691/b052691.txt">Table of n, a(n) for n = 0..375</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=639">Encyclopedia of Combinatorial Structures 639</a>

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

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

%F a(n) = (n!/23)*Sum_{alpha=RootOf(-1+2*Z-Z^2+Z^3)} (1 + 6*alpha + 3*alpha^2)*_alpha^(-1-n).

%F a(n) = n!*A005251(n+1). - _R. J. Mathar_, Nov 27 2011

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

%t With[{nn=20},CoefficientList[Series[(1-x)/(1-2x+x^2-x^3),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, May 09 2018 *)

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x)/(1-2*x+x^2-x^3) ))); // _G. C. Greubel_, Jun 02 2022

%o (SageMath) [factorial(n)*sum(binomial(n-j, 2*j) for j in (0..n//3)) for n in (0..40)] # _G. C. Greubel_, Jun 02 2022

%Y Cf. A000142, A005251.

%K easy,nonn

%O 0,3

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