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

2, 1, 4, 6, 48, 120, 1440, 5040, 80640, 362880, 7257600, 39916800, 958003200, 6227020800, 174356582400, 1307674368000, 41845579776000, 355687428096000, 12804747411456000, 121645100408832000, 4865804016353280000, 51090942171709440000, 2248001455555215360000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 508

Crossrefs

Cf. A000142.

Programs

Maple

spec := [S,{S=Union(Sequence(Z), Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a:=n->n!+sum((-1)^k*n!, k=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, Mar 25 2008

Mathematica

a[n_] := If[OddQ[n], n!, 2*n!]; Array[a, 20, 0] (* Amiram Eldar, Jul 06 2022 *)

PARI

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

Formula

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

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

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

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

From Amiram Eldar, Jul 06 2022: (Start)

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

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