A052616 Expansion of e.g.f. (3+2*x)/(1-x^2).

3, 2, 6, 12, 72, 240, 2160, 10080, 120960, 725760, 10886400, 79833600, 1437004800, 12454041600, 261534873600, 2615348736000, 62768369664000, 711374856192000, 19207121117184000, 243290200817664000, 7298706024529920000, 102181884343418880000, 3372002183332823040000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 561.

Crossrefs

Cf. A176059.

Programs

Maple

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

Mathematica

With[{nn=30},CoefficientList[Series[(3+2x)/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 12 2021 *)

Formula

E.g.f.: (2*x+3)/(1-x^2).

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

a(n) = Sum(1/2*(3*_alpha+2)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^2))*n!.

a(n) = 3n! if n is even, 2n! otherwise.

a(n) = n!*A176059(n). - R. J. Mathar, Jun 03 2022

Sum_{n>=0} 1/a(n) = (5*e^2-1)/(12*e) = cosh(1)/3 + sinh(1)/2. - Amiram Eldar, Feb 02 2023