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A052589 a(n) = (2^n - 1)*n!.

%I A052589 #34 Jun 03 2022 18:08:09
%S A052589 0,1,6,42,360,3720,45360,640080,10281600,185431680,3712262400,
%T A052589 81709689600,1961511552000,51005527372800,1428241944729600,
%U A052589 42848566016256000,1371175035310080000,46620306887970816000,1678337450340655104000,63776944758045302784000
%N A052589 a(n) = (2^n - 1)*n!.
%H A052589 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=534">Encyclopedia of Combinatorial Structures 534</a>
%F A052589 E.g.f.: x / ((1-2*x) * (1-x)).
%F A052589 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 A052589 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 A052589 From _Michael Somos_, Jul 22 2017: (Start)
%F A052589 If A(x) = Sum_{k>0} x^k / a(k), then A(2*x) = A(x) + e^x - 1.
%F A052589 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 A052589 spec := [S,{S=Prod(Z,Sequence(Z),Sequence(Union(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052589 Table[(2^n-1)n!,{n,0,20}] (* _Harvey P. Dale_, Jul 18 2015 *)
%o A052589 (PARI) {a(n) = if( n<0, 0, (2^n - 1)*n!)}; /* _Michael Somos_, Jul 22 2017 */
%Y A052589 Cf. A000165.
%K A052589 easy,nonn
%O A052589 0,3
%A A052589 encyclopedia(AT)pommard.inria.fr, Jan 25 2000