cp's OEIS Frontend

A052674 Expansion of e.g.f. (1-x)/(1-3*x-2*x^2+2*x^3).

%I A052674 #20 Jan 05 2025 01:13:16
%S A052674 1,2,16,156,2160,36720,753120,17992800,491500800,15102339840,
%T A052674 515630707200,19365156518400,793401964185600,35214960849868800,
%U A052674 1683239666985676800,86204093846846976000,4709107007890661376000,273324248772505362432000,16797372435596048744448000
%N A052674 Expansion of e.g.f. (1-x)/(1-3*x-2*x^2+2*x^3).
%H A052674 G. C. Greubel, <a href="/A052674/b052674.txt">Table of n, a(n) for n = 0..350</a>
%H A052674 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=622">Encyclopedia of Combinatorial Structures 622</a>
%F A052674 E.g.f.: (1 - x)/((1 + x)*(1 - 4*x + 2*x^2)).
%F A052674 Recurrence: a(0)=1, a(1)=2, a(2)=16, a(n) = 3*n*a(n-1) + 2*n*(n-1)*a(n-2) - 2*n*(n-1)*(n-2)*a(n-3).
%F A052674 a(n) = (n!/98)*Sum_{alpha=RootOf(1 -3*Z -2*Z^2 +2*Z^3)} (13 + 25*alpha - 16*alpha^2)*alpha^(-1-n).
%F A052674 a(n) = n!*A052543(n). - _R. J. Mathar_, Nov 27 2011
%F A052674 a(n) = (n!/7)*(2*(-1)^n + 2^(n/2)*( 5*ChebyshevU(n, sqrt(2)) - 2*sqrt(2)*ChebyshevU(n-1, sqrt(2)) )). - _G. C. Greubel_, Jun 12 2022
%p A052674 spec := [S,{S=Sequence(Prod(Union(Z,Z),Union(Z,Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052674 Table[(n!/7)*(2*(-1)^n + 2^(n/2)*(5*ChebyshevU[n, Sqrt[2]] - 2*Sqrt[2]*ChebyshevU[n - 1, Sqrt[2]])), {n, 0, 30}] (* _G. C. Greubel_, Jun 12 2022 *)
%o A052674 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1-x)/((1+x)*(1-4*x+2*x^2)) ))); // _G. C. Greubel_, Jun 12 2022
%o A052674 (SageMath) [factorial(n)*(2^(n/2)*(5*chebyshev_U(n,sqrt(2)) - 2*sqrt(2)*chebyshev_U(n-1, sqrt(2))) + 2*(-1)^n)/7 for n in (0..30)] # _G. C. Greubel_, Jun 12 2022
%Y A052674 Cf. A000142, A052543.
%K A052674 easy,nonn
%O A052674 0,2
%A A052674 encyclopedia(AT)pommard.inria.fr, Jan 25 2000