%I #19 Jun 11 2022 05:24:10
%S 1,3,22,246,3672,68520,1534320,40083120,1196737920,40196580480,
%T 1500156806400,61585275628800,2758072531737600,133812468652262400,
%U 6991529043750451200,391391124208051968000,23371064978815217664000
%N Expansion of e.g.f. (1-x)/(1-4*x+x^2).
%H G. C. Greubel, <a href="/A052677/b052677.txt">Table of n, a(n) for n = 0..365</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=625">Encyclopedia of Combinatorial Structures 625</a>
%F E.g.f.: (1-x)/(1-4*x+x^2).
%F Recurrence: a(0)=1, a(1)=3, a(n+2) = 4*(n+2)*a(n+1) - (n+2)*(n+1)*a(n).
%F a(n) = (n!/6)*Sum_{alpha=RootOf(1 -4*Z +Z^2)} (1 + alpha)*alpha^(-1-n).
%F a(n) = n!*A079935(n). - _R. J. Mathar_, Nov 27 2011
%F a(n) = (-1)^n * n! * ChebyshevU(2*n, i/sqrt(2)). - _G. C. Greubel_, Jun 11 2022
%p spec := [S,{S=Sequence(Union(Z,Prod(Sequence(Z),Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t With[{nn=20},CoefficientList[Series[(1-x)/(1-4x+x^2),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 28 2016 *)
%o (Magma) [n le 2 select 3^(n-1) else 4*(n-1)*Self(n-1) - (n-1)*(n-2)*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Jun 11 2022
%o (SageMath) [factorial(n)*sum( binomial(2*n-k, k)*2^(n-k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 11 2022
%Y Cf. A000142, A079935.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000