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A052566 Expansion of e.g.f. (2 + x)/(1 - x^2).

%I A052566 #25 Jul 06 2022 06:58:43
%S A052566 2,1,4,6,48,120,1440,5040,80640,362880,7257600,39916800,958003200,
%T A052566 6227020800,174356582400,1307674368000,41845579776000,355687428096000,
%U A052566 12804747411456000,121645100408832000,4865804016353280000,51090942171709440000,2248001455555215360000
%N A052566 Expansion of e.g.f. (2 + x)/(1 - x^2).
%H A052566 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=508">Encyclopedia of Combinatorial Structures 508</a>
%F A052566 Recurrence: {a(1)=1, a(0)=2, (-2 - n^2 - 3*n)*a(n) + a(n+2) = 0}.
%F A052566 Sum((1/2)*(1 + 2*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1 + _Z^2))*n!.
%F A052566 E.g.f.: (x+2)/(1-x^2).
%F A052566 a(n) = 2n! if n is even, n! if odd.
%F A052566 From _Amiram Eldar_, Jul 06 2022: (Start)
%F A052566 Sum_{n>=0} 1/a(n) = sinh(1) + cosh(1)/2.
%F A052566 Sum_{n>=0} (-1)^(n+1)/a(n) = sinh(1) - cosh(1)/2. (End)
%p A052566 spec := [S,{S=Union(Sequence(Z), Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052566 a:=n->n!+sum((-1)^k*n!, k=0..n): seq(a(n), n=0..19); # _Zerinvary Lajos_, Mar 25 2008
%t A052566 a[n_] := If[OddQ[n], n!, 2*n!]; Array[a, 20, 0] (* _Amiram Eldar_, Jul 06 2022 *)
%o A052566 (PARI) a(n)=if(n<0,0,n!*polcoeff((x+2)/(1-x^2)+x*O(x^n),n))
%Y A052566 Cf. A000142.
%K A052566 easy,nonn
%O A052566 0,1
%A A052566 encyclopedia(AT)pommard.inria.fr, Jan 25 2000