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A052765 Expansion of e.g.f.: -x^2*(log(1-x))^3.

%I A052765 #22 Aug 08 2020 16:08:36
%S A052765 0,0,0,0,0,120,1080,8820,75600,701568,7091280,77961840,928778400,
%T A052765 11937347136,164802429792,2433765035520,38299272560640,
%U A052765 639999894048768,11320441140625920,211340086405770240,4153253573744179200,85710868976433254400,1853386172505676892160
%N A052765 Expansion of e.g.f.: -x^2*(log(1-x))^3.
%C A052765 Previous name was: A simple grammar.
%H A052765 G. C. Greubel, <a href="/A052765/b052765.txt">Table of n, a(n) for n = 0..448</a>
%H A052765 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=722">Encyclopedia of Combinatorial Structures 722</a>
%F A052765 E.g.f.: x^2*log(-1/(-1+x))^3.
%F A052765 Recurrence: a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=120, (16*n-48*n^2-4*n^3-n^6+13*n^4+48)*a(n) + (n^2+61*n-26*n^3+3*n^4+3*n^5-42)*a(n+1) + (-9*n+15*n^2-3*n^3-3*n^4)*a(n+2) + (n^3-n)*a(n+3) = 0.
%F A052765 a(n) ~ (n-1)! * (3*log(n)^2 + 6*gamma*log(n) - Pi^2/2 + 3*gamma^2), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 01 2013
%F A052765 a(n) = n*A052758(n-1) = 3!*n*(n-1)*abs(Stirling1(n-2,3)) for n >= 2. - _Andrew Howroyd_, Aug 08 2020
%p A052765 spec := [S,{B=Cycle(Z),S=Prod(Z,Z,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052765 CoefficientList[Series[-x^2*(Log[1-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 01 2013 *)
%o A052765 (PARI) x='x+O('x^30); concat(vector(5), Vec(serlaplace(x^2*log(-1/(-1+x))^3))) \\ _G. C. Greubel_, Sep 05 2018
%o A052765 (PARI) a(n)={if(n>=2, 3!*n*(n-1)*abs(stirling(n-2,3,1)), 0)} \\ _Andrew Howroyd_, Aug 08 2020
%Y A052765 Cf. A052754, A052758.
%K A052765 easy,nonn
%O A052765 0,6
%A A052765 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052765 New name using e.g.f., _Vaclav Kotesovec_, Oct 01 2013