A052566 - OEIS

%I #25 Jul 06 2022 06:58:43

%S 2,1,4,6,48,120,1440,5040,80640,362880,7257600,39916800,958003200,

%T 6227020800,174356582400,1307674368000,41845579776000,355687428096000,

%U 12804747411456000,121645100408832000,4865804016353280000,51090942171709440000,2248001455555215360000

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

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=508">Encyclopedia of Combinatorial Structures 508</a>

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

%F Sum((1/2)*(1 + 2*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1 + _Z^2))*n!.

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

%F a(n) = 2n! if n is even, n! if odd.

%F From _Amiram Eldar_, Jul 06 2022: (Start)

%F Sum_{n>=0} 1/a(n) = sinh(1) + cosh(1)/2.

%F Sum_{n>=0} (-1)^(n+1)/a(n) = sinh(1) - cosh(1)/2. (End)

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

%p a:=n->n!+sum((-1)^k*n!, k=0..n): seq(a(n), n=0..19); # _Zerinvary Lajos_, Mar 25 2008

%t a[n_] := If[OddQ[n], n!, 2*n!]; Array[a, 20, 0] (* _Amiram Eldar_, Jul 06 2022 *)

%o (PARI) a(n)=if(n<0,0,n!*polcoeff((x+2)/(1-x^2)+x*O(x^n),n))

%Y Cf. A000142.

%K easy,nonn

%O 0,1

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