A052589 - OEIS

%I #34 Jun 03 2022 18:08:09

%S 0,1,6,42,360,3720,45360,640080,10281600,185431680,3712262400,

%T 81709689600,1961511552000,51005527372800,1428241944729600,

%U 42848566016256000,1371175035310080000,46620306887970816000,1678337450340655104000,63776944758045302784000

%N a(n) = (2^n - 1)*n!.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=534">Encyclopedia of Combinatorial Structures 534</a>

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

%F D-finite with Recurrence: {a(1)=1, a(0)=0, (2*n^2 + 6*n + 4)*a(n) + (-6 - 3*n)*a(n+1) + a(n+2) = 0}.

%F G.f.: -G(0) where G(k) = 1 - 2^k/(1 - x*(k+1)/(x*(k+1) - 2^k/G(k+1) )), (continued fraction). - _Sergei N. Gladkovskii_, Dec 06 2012

%F From _Michael Somos_, Jul 22 2017: (Start)

%F If A(x) = Sum_{k>0} x^k / a(k), then A(2*x) = A(x) + e^x - 1.

%F 0 = +a(n)*(+1104*a(n+3) -792*a(n+4) +136*a(n+5) -6*a(n+6)) +a(n+1)*(+828*a(n+3) -435*a(n+4) +39*a(n+5)) + a(n+2)*(+299*a(n+3) -102*a(n+4)) +a(n+3)*(+69*a(n+3)) for n>=0. (End)

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

%t Table[(2^n-1)n!,{n,0,20}] (* _Harvey P. Dale_, Jul 18 2015 *)

%o (PARI) {a(n) = if( n<0, 0, (2^n - 1)*n!)}; /* _Michael Somos_, Jul 22 2017 */

%Y Cf. A000165.

%K easy,nonn

%O 0,3

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