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A052591 Expansion of e.g.f. x/((1-x)(1-x^2)).

%I A052591 #43 Jan 22 2023 08:41:47
%S A052591 0,1,2,12,48,360,2160,20160,161280,1814400,18144000,239500800,
%T A052591 2874009600,43589145600,610248038400,10461394944000,167382319104000,
%U A052591 3201186852864000,57621363351552000,1216451004088320000,24329020081766400000,562000363888803840000
%N A052591 Expansion of e.g.f. x/((1-x)(1-x^2)).
%C A052591 Stirling transform of 2*a(n) = [2,4,24,96,...] is A052841(n+1) = [2,6,38,270,...]. - _Michael Somos_, Mar 04 2004
%C A052591 a(n) is the number of even fixed points in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 12'3, 132, 312, 213, 231, and 32'1, the even fixed points being marked. - _Emeric Deutsch_, Jul 18 2009
%H A052591 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=536">Encyclopedia of Combinatorial Structures 536</a>.
%F A052591 Recurrence: {a(1)=1, a(0)=0, (-n^3 - 5*n^2 - 8*n - 4)*a(n) + (-2-n)*a(n+1) + (n+1)*a(n+2) = 0}.
%F A052591 a(n) = ((1/4)*(-1)^(1-n) + (1/2)*n + 1/4)*n!.
%F A052591 E.g.f.: x/((1-x)*(1-x^2)).
%F A052591 From _Emeric Deutsch_, Jul 18 2009: (Start)
%F A052591 a(n) = (n+1)!/2 if n is odd; a(n) = n!*n/2 if n is even.
%F A052591 a(n) = (n+1)! - A052558(n). (End)
%F A052591 a(n) = n!*A008619(n-1), n > 1. - _R. J. Mathar_, Nov 27 2011
%F A052591 Sum_{n>=1} 1/a(n) = 2*(CoshIntegral(1) + cosh(1) - gamma - 1) = 2*(A099284 + A073743 - A001620 - 1). - _Amiram Eldar_, Jan 22 2023
%p A052591 spec := [S,{S=Prod(Z,Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052591 G(x):=x/(1-x)/(1-x^2): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # _Zerinvary Lajos_, Apr 03 2009
%o A052591 (PARI) a(n)=if(n<0,0,n!*polcoeff(x/(1-x)/(1-x^2)+x*O(x^n),n))
%Y A052591 Cf. A008619, A052841.
%Y A052591 Cf. A052558. - _Emeric Deutsch_, Jul 18 2009
%Y A052591 Cf. A001620, A073743, A099284.
%K A052591 easy,nonn
%O A052591 0,3
%A A052591 encyclopedia(AT)pommard.inria.fr, Jan 25 2000