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A052584 E.g.f. (2-4x+x^2)/((1-x)(1-2x)). 3

%I

%S 2,2,6,30,216,2040,23760,327600,5201280,93260160,1861574400,

%T 40914720000,981474278400,25512104217600,714251739801600,

%U 21426244519680000,685618901839872000,23310686975127552000,839178328730886144000,31888654846673264640000,1275543760964922408960000

%N E.g.f. (2-4x+x^2)/((1-x)(1-2x)).

%H Vincenzo Librandi, <a href="/A052584/b052584.txt">Table of n, a(n) for n = 0..400</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=529">Encyclopedia of Combinatorial Structures 529</a>

%F E.g.f.: (2-4*x+x^2)/(-1+2*x)/(-1+x).

%F 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 a(n) = (1+2^(n-1))*n!, n > 0; see A000051.

%F From _Peter Luschny_, Apr 20 2009: (Start)

%F 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 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 spec := [S,{S=Union(Sequence(Prod(Z,Sequence(Z))),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p 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 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 (MAGMA) [2] cat [(1+2^(n-1))*Factorial(n): n in [1..20]]; // _Vincenzo Librandi_, Aug 11 2017

%K easy,nonn

%O 0,1

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

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Last modified May 15 17:50 EDT 2021. Contains 343920 sequences. (Running on oeis4.)