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A052668 Expansion of e.g.f. 1/(1 - 3*x - x^3).

%I A052668 #17 Sep 03 2022 09:45:31
%S A052668 1,3,18,168,2088,32400,603360,13109040,325503360,9092684160,
%T A052668 282219033600,9635476435200,358879494758400,14480588157235200,
%U A052668 629228583138355200,29295027261916416000,1454816084780298240000
%N A052668 Expansion of e.g.f. 1/(1 - 3*x - x^3).
%H A052668 G. C. Greubel, <a href="/A052668/b052668.txt">Table of n, a(n) for n = 0..375</a>
%H A052668 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=616">Encyclopedia of Combinatorial Structures 616</a>
%F A052668 E.g.f.: 1/(1-3*x-x^3).
%F A052668 a(n) = 3*n*a(n-1) + n*(n-1)*(n-2)*a(n-3), a(0)=1, a(1)=3, a(2)=18.
%F A052668 a(n) = (n!/15) * Sum_{alpha=RootOf(-1+3*_Z+_Z^3)} (4 + alpha + 2*alpha^2) * alpha^(-1-n).
%F A052668 a(n) = n!*A052541(n). - _R. J. Mathar_, Nov 27 2011
%p A052668 spec := [S,{S=Sequence(Union(Z,Z,Z,Prod(Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052668 a[n_]:= a[n]= If[n<3, 3^n*n!, 3*n*a[n-1] + n*(n-1)*(n-2)*a[n-3]];
%t A052668 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Sep 03 2022 *)
%o A052668 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( 1/(1-3*x-x^3) ))); // _G. C. Greubel_, Sep 03 2022
%o A052668 (SageMath)
%o A052668 def A052668_list(prec):
%o A052668     P.<x> = PowerSeriesRing(QQ, prec)
%o A052668     return P( 1/(1-3*x-x^3) ).egf_to_ogf().list()
%o A052668 A052668_list(40) # _G. C. Greubel_, Sep 03 2022
%Y A052668 Cf. A052541.
%K A052668 easy,nonn
%O A052668 0,2
%A A052668 encyclopedia(AT)pommard.inria.fr, Jan 25 2000