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%I #17 Jun 15 2022 01:51:06
%S 1,1,8,66,840,12960,242640,5286960,131765760,3693755520,115058361600,
%T 3942342835200,147360531225600,5967185903078400,260221271108198400,
%U 12158477739023616000,605960547270414336000,32087688283562655744000
%N Expansion of e.g.f. (1-2*x)/(1-3*x-x^2+2*x^3).
%H G. C. Greubel, <a href="/A052669/b052669.txt">Table of n, a(n) for n = 0..350</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=617">Encyclopedia of Combinatorial Structures 617</a>
%F E.g.f.: (1 - 2*x)/(1 - 3*x - x^2 + 2*x^3).
%F Recurrence: a(0)=1, a(1)=1, a(2)=8, a(n) = 3*n*a(n-1) + n*(n-1)*a(n-2) - 2*n*(n-1)*(n-2)*a(n-3).
%F a(n) = (n!/229)*Sum_{alpha=RootOf(1 - 3*Z - Z^2 + 2*Z^3)} (5 + 74*alpha - 16*alpha^2)*alpha^(-1-n).
%F a(n) = n!*A052550(n). - _R. J. Mathar_, Nov 27 2011
%p spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Union(Z,Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t b[n_]:= b[n]= If[n<3, 1+3*Floor[n/2], 3*b[n-1] +b[n-2] -2*b[n-3]];
%t A052669[n_] := n!*b[n]; (* b = A052550 *)
%t Table[A052669[n], {n, 0, 40}] (* _G. C. Greubel_, Jun 14 2022 *)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-2*x)/(1-3*x-x^2+2*x^3) ))); // _G. C. Greubel_, Jun 14 2022
%o (SageMath)
%o @CachedFunction
%o def b(n): # b = A052550
%o if (n<3): return 1 + 3*(n//2)
%o else: return 3*b(n-1) +b(n-2) -2*b(n-3)
%o def A052669(n): return factorial(n)*b(n)
%o [A052669(n) for n in (0..40)] # _G. C. Greubel_, Jun 14 2022
%Y Cf. A000142, A052550.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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