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

%I #33 Mar 15 2023 11:04:48

%S 0,0,0,0,24,240,1440,6720,26880,96768,322560,1013760,3041280,8785920,

%T 24600576,67092480,178913280,467927040,1203240960,3048210432,

%U 7620526080,18827182080,46022000640,111421685760,267412045824,636695347200,1504916275200

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

%C 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.

%H Vincenzo Librandi, <a href="/A052796/b052796.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=753">Encyclopedia of Combinatorial Structures 753</a>.

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

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

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

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

%F a(n) = (n-3)*(n-2)*(n-1)*n * 2^(n-4). - _Vaclav Kotesovec_, Nov 27 2012

%F From _Amiram Eldar_, Jan 09 2022: (Start)

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

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

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

%t Range[0, 30]!* CoefficientList[Series[Exp[x]^2 * x^4,{x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 06 2012 *)

%o (Magma) [(n-3)*(n-2)*(n-1)*n * 2^(n-4): n in [0..30]]; // _Vincenzo Librandi_, Dec 06 2012

%Y Cf. A090802.

%K easy,nonn

%O 0,5

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

%E More terms from _Vincenzo Librandi_, Dec 06 2012