A052796 Expansion of e.g.f. x^4*exp(x)^2.

0, 0, 0, 0, 24, 240, 1440, 6720, 26880, 96768, 322560, 1013760, 3041280, 8785920, 24600576, 67092480, 178913280, 467927040, 1203240960, 3048210432, 7620526080, 18827182080, 46022000640, 111421685760, 267412045824, 636695347200, 1504916275200

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

a(n) is the number of ways that n people can form two distinct committees and then choose a president and vice president for each committee.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 753.

Crossrefs

Cf. A090802.

Programs

Magma

[(n-3)*(n-2)*(n-1)*n * 2^(n-4): n in [0..30]]; // Vincenzo Librandi, Dec 06 2012

Maple

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

Mathematica

Range[0, 30]!* CoefficientList[Series[Exp[x]^2 * x^4,{x, 0, 30}], x] (* Vincenzo Librandi, Dec 06 2012 *)

Formula

E.g.f.: x^4*exp(x)^2.

a(n) = A090802(n, 4).

Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-2*n-2)*a(n)+(n-3)*a(n+1)}.

O.g.f.: -24*x^4/(2*x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

a(n) = (n-3)*(n-2)*(n-1)*n * 2^(n-4). - Vaclav Kotesovec, Nov 27 2012

From Amiram Eldar, Jan 09 2022: (Start)

Sum_{n>=4} 1/a(n) = 5/18 - log(2)/3.

Sum_{n>=4} (-1)^n/a(n) = 9*log(3/2) - 65/18. (End)

Extensions

More terms from Vincenzo Librandi, Dec 06 2012