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A052584 Expansion of e.g.f. (2 - 4*x + x^2)/((1 - x)*(1 - 2*x)).

%I A052584 #29 Oct 27 2023 12:59:16
%S A052584 2,2,6,30,216,2040,23760,327600,5201280,93260160,1861574400,
%T A052584 40914720000,981474278400,25512104217600,714251739801600,
%U A052584 21426244519680000,685618901839872000,23310686975127552000,839178328730886144000,31888654846673264640000,1275543760964922408960000
%N A052584 Expansion of e.g.f. (2 - 4*x + x^2)/((1 - x)*(1 - 2*x)).
%H A052584 Vincenzo Librandi, <a href="/A052584/b052584.txt">Table of n, a(n) for n = 0..400</a>
%H A052584 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=529">Encyclopedia of Combinatorial Structures 529</a>
%F A052584 Recurrence: {a(1)=2, a(2)=6, a(0)=2, (2*n^2 + 6*n + 4)*a(n) + (-6-3*n)*a(n+1) + a(n+2) = 0}.
%F A052584 a(n) = (1+2^(n-1))*n!, n > 0; see A000051.
%F A052584 From _Peter Luschny_, Apr 20 2009: (Start)
%F A052584 A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
%F A052584 a(n) = Sum_{k=0..n-1} n!*C(n-1,k)*B_{n-k-1}*2^(k+1)/(k+1). (See also A000051.) (End)
%p A052584 spec := [S,{S=Union(Sequence(Prod(Z,Sequence(Z))),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052584 a := proc(n) if n = 0 then 2 else add(n!*binomial(n-1,k)*bernoulli(n-k-1,1)*2^(k+ 1)/(k+1),k=0..n-1) fi end: # _Peter Luschny_, Apr 20 2009
%t A052584 With[{nn=25}, CoefficientList[Series[(2 - 4 x + x^2) / (-1 + 2 x) / (-1 + x), {x, 0, nn}], x] Range[0, nn]!] (* _Vincenzo Librandi_, Aug 11 2017 *)
%o A052584 (Magma) [2] cat [(1+2^(n-1))*Factorial(n): n in [1..20]]; // _Vincenzo Librandi_, Aug 11 2017
%o A052584 (PARI) Vec(serlaplace((2-4*x+x^2)/((1-x)*(1-2*x)) + O(x^20))) \\ _Andrew Howroyd_, Oct 26 2023
%Y A052584 Cf. A000051, A027641, A027642.
%K A052584 easy,nonn
%O A052584 0,1
%A A052584 encyclopedia(AT)pommard.inria.fr, Jan 25 2000