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%I #23 Aug 09 2020 00:32:24
%S 0,0,0,0,0,120,1080,6300,30240,130032,521640,1996500,7389360,26676936,
%T 94486392,329647500,1136116800,3876164832,13112135496,44031456900,
%U 146920942800,487489214520,1609441068312,5289755245500,17315399138400,56470807803600,183546483143400
%N E.g.f.: x^2*(exp(x)-1)^3.
%C Previous name was: A simple grammar.
%H Andrew Howroyd, <a href="/A052777/b052777.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=734">Encyclopedia of Combinatorial Structures 734</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (18,-141,630,-1767,3222,-3815,2826,-1188,216).
%F E.g.f.: x^2*exp(x)^3-3*x^2*exp(x)^2+3*exp(x)*x^2-x^2.
%F Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (-36*n^2-66*n-6*n^3-36)*a(n)+(11*n+11*n^3+44*n^2-66)*a(n+1)+(-12*n^2+18*n-6*n^3)*a(n+2)+(n^3-n)*a(n+3), a(5)=120}.
%F For n>2, a(n) = n*(n-1)*(3^(n-2) - 3*2^(n-2) + 3). - _Vaclav Kotesovec_, Oct 01 2013
%F a(n) = n*A052761(n-1) = 3!*n*(n-1)*Stirling2(n-2,3) for n >= 2. - _Andrew Howroyd_, Aug 08 2020
%p spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,Z,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20); # end of program
%p seq(6*(n^2-n)*combinat[stirling2](n-2,3), n=0..20); # _Mark van Hoeij_, May 29 2013
%t CoefficientList[Series[x^2*(E^x-1)^3, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 01 2013 *)
%o (PARI) x='x+O('x^66); concat([0,0,0,0,0], Vec( serlaplace( x^2*exp(x)^3-3*x^2*exp(x)^2+3*exp(x)*x^2-x^2))) \\ _Joerg Arndt_, May 29 2013
%o (PARI) a(n)={if(n>=2, 3!*n*(n-1)*stirling(n-2,3,2), 0)} \\ _Andrew Howroyd_, Aug 08 2020
%Y Cf. A052760, A052761.
%K easy,nonn
%O 0,6
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name using e.g.f., _Vaclav Kotesovec_, Oct 01 2013
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