%I #24 Aug 12 2024 13:24:12
%S 0,1,2,6,48,240,1440,15120,120960,1088640,14515200,159667200,
%T 1916006400,31135104000,435891456000,6538371840000,125536739328000,
%U 2134124568576000,38414242234368000,851515702861824000
%N Expansion of e.g.f. x/((1-x)*(1-x^3)).
%H G. C. Greubel, <a href="/A052688/b052688.txt">Table of n, a(n) for n = 0..350</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=636">Encyclopedia of Combinatorial Structures 636</a>
%F Recurrence: a(0)=0, a(1)=1, a(2)=2, (n+2)*a(n+3) = (n+3)*a(n+2) + (n+2)*(n+3)*a(n+1) + (n+1)*(n+2)*(n+3)^2*a(n).
%F a(n) = (n!/3)*(n + 1 - (1/3)*Sum_{alpha=RootOf(Z^2 + Z + 1)} (1 + 2*alpha)*_alpha^(-1-n)).
%F a(n) = n!*A002264(n+2). - _R. J. Mathar_, Nov 27 2011
%p spec := [S,{S=Prod(Z,Sequence(Z),Sequence(Prod(Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p G(x):=x/(1-x)/(1-x^3): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # _Zerinvary Lajos_, Apr 03 2009
%t With[{nn=20},CoefficientList[Series[x/((1-x)(1-x^3)),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 24 2018 *)
%t Table[n!*Floor[(n+2)/3], {n,0,40}] (* _G. C. Greubel_, Jun 02 2022 *)
%o (SageMath) [factorial(n)*((n+2)//3) for n in (0..40)] # _G. C. Greubel_, Jun 02 2022
%Y Cf. A000142, A002264.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000