A052657 Expansion of e.g.f. x^2/((1-x)^2*(1+x)).

0, 0, 2, 6, 48, 240, 2160, 15120, 161280, 1451520, 18144000, 199584000, 2874009600, 37362124800, 610248038400, 9153720576000, 167382319104000, 2845499424768000, 57621363351552000, 1094805903679488000, 24329020081766400000, 510909421717094400000, 12364008005553684480000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Stirling transform of -(-1)^n*a(n-1) = [0, 0, 2, -6, 48, -240, ...] is A052841(n-1) = [0, 0, 2, 6, 38, 270, ...]. - Michael Somos, Mar 04 2004

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 604.

Crossrefs

Cf. A004526, A052841.

Cf. A001620, A073742, A099284.

Programs

Maple

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

Mathematica

a[n_] := Floor[n/2] * n!; Array[a, 25, 0] (* Amiram Eldar, Jan 22 2023 *)

PARI

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

PARI

a(n)=n!*(n\2); \\ Joerg Arndt, Jan 22 2023

Formula

a(0)=0, a(1)=0, a(2)=2, n*a(n+2) = (n+2)*a(n+1) + (n^3 + 4*n^2 + 5*n + 2)*a(n).

a(n) = (2*n-1+(-1)^n)*n!/4 = n!*floor(n/2) = n!*A004526(n).

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

Sum_{n>=2} 1/a(n) = 4*CoshIntegral(1) - 4*gamma - 2*sinh(1) + 2 = 4*A099284 - 4*A001620 - 2*A073742 + 2. - Amiram Eldar, Jan 22 2023