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

0,1

Links

Table of n, a(n) for n=0..22.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 508

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)

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 *)

Prog

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

Crossrefs

Cf. A000142.

Sequence in context: A328491 A328357 A341696 * A234946 A223092 A071948

Adjacent sequences: A052563 A052564 A052565 * A052567 A052568 A052569

Keyword

easy,nonn

Author

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

Status

approved