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A052644 Expansion of e.g.f. (1+3x-3x^2)/(1-x)^2.

%I A052644 #31 Jan 05 2025 00:45:38
%S A052644 1,5,12,42,192,1080,7200,55440,483840,4717440,50803200,598752000,
%T A052644 7664025600,105859353600,1569209241600,24845812992000,418455797760000,
%U A052644 7469435990016000,140852221526016000
%N A052644 Expansion of e.g.f. (1+3x-3x^2)/(1-x)^2.
%H A052644 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=590">Encyclopedia of Combinatorial Structures 590</a>
%F A052644 E.g.f.: -(-3*x+3*x^2-1)/(-1+x)^2
%F A052644 Recurrence: {a(0)=1, a(1)=5, (-n^2-6*n-5)*a(n)+(n+4)*a(n+1)=0, a(2)=12}.
%F A052644 a(n) = (n+4)*n!, n>0.
%F A052644 G.f.: G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+4)/G(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Oct 16 2012
%F A052644 From _Amiram Eldar_, Nov 06 2020: (Start)
%F A052644 Sum_{n>=0} 1/a(n) = 27/4- 2*e.
%F A052644 Sum_{n>=0} (-1)^n/a(n) = 27/4 - 16/e. (End)
%p A052644 spec := [S,{S=Prod(Sequence(Z),Union(Z,Z,Z,Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052644 With[{nn=20},CoefficientList[Series[(1+3x-3x^2)/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 06 2014 *)
%Y A052644 Cf. sequences with formula (n + k)*n! listed in A282466.
%K A052644 easy,nonn
%O A052644 0,2
%A A052644 encyclopedia(AT)pommard.inria.fr, Jan 25 2000