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%I #28 Sep 08 2022 08:44:59
%S 0,0,0,0,24,120,420,1260,3472,9072,22860,56100,134904,319176,745108,
%T 1719900,3931680,8912352,20053404,44825940,99613960,220200120,
%U 484441188,1061157900,2315254704,5033163600,10905189100,23555209860,50734299672,108984793512
%N Expansion of e.g.f.: x^2*(exp(x)-1)^2.
%C Original name: a simple grammar.
%H Vincenzo Librandi, <a href="/A052760/b052760.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=717">Encyclopedia of Combinatorial Structures 717</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,63,-66,36,-8).
%F E.g.f.: x^2*exp(x)^2-2*exp(x)*x^2+x^2.
%F Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (2*n^2+6*n+4)*a(n)+(6-3*n^2-3*n)*a(n+1)+(n^2-n)*a(n+2)}.
%F For n>=3, a(n) = n*(n-1)*(2^n-8)/4. - _Vaclav Kotesovec_, Nov 27 2012
%F a(n) = n*A052749(n-1) = 2*n*(n-1)*Stirling2(n-2,2) for n >= 2. - _Andrew Howroyd_, Aug 08 2020
%p spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,Z,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t Part[#, Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0] #)&@CoefficientList[Series[x^2 Exp[x]^2 - 2 Exp[x] x^2 + x^2, {x, 0, 30}], x]//ExpandAll (* _Vincenzo Librandi_, May 05 2013 *)
%o (Magma) [0,0,0] cat [n*(n-1)*(2^n-8)/4: n in [3..30]]; // _Vincenzo Librandi_, May 05 2013
%o (PARI) a(n) = if(n<4, 0, n*(n-1)*(2^n-8)/4); \\ _Joerg Arndt_, May 06 2013
%Y Cf. A052749.
%K easy,nonn
%O 0,5
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _Vincenzo Librandi_, May 05 2013
%E Name changed by _Andrew Howroyd_, Aug 08 2020
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