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A052799 Expansion of e.g.f.: x^4*(log(1-x))^2.

%I A052799 #21 Aug 08 2020 16:09:03
%S A052799 0,0,0,0,0,0,720,5040,36960,302400,2761920,27941760,310495680,
%T A052799 3760922880,49324923648,696388492800,10530709862400,169811234611200,
%U A052799 2908629247795200,52738216760033280,1009115747652096000
%N A052799 Expansion of e.g.f.: x^4*(log(1-x))^2.
%C A052799 Previous name was: A simple grammar.
%H A052799 G. C. Greubel, <a href="/A052799/b052799.txt">Table of n, a(n) for n = 0..449</a>
%H A052799 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=757">Encyclopedia of Combinatorial Structures 757</a>
%F A052799 E.g.f.: x^4*log(-1/(-1+x))^2.
%F A052799 Recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (-5*n^3+n^4+32*n-6*n^2+32)*a(n)+(9*n^2+5*n-2*n^3-42)*a(n+1)+(n^2-5*n+6)*a(n+2) = 0.
%F A052799 a(n) ~ (n-1)! * 2*(log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 01 2013
%F A052799 a(n) = n*A052766(n-1) = 2*4!*binomial(n,4)*abs(Stirling1(n-4,2)) for n >= 4. - _Andrew Howroyd_, Aug 08 2020
%p A052799 spec := [S,{B=Cycle(Z),S=Prod(Z,Z,Z,Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052799 CoefficientList[Series[x^4*(Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 01 2013 *)
%o A052799 (PARI) x='x+O('x^30); concat(vector(6), Vec(serlaplace(x^4*(log(1-x))^2))) \\ _G. C. Greubel_, Sep 05 2018
%o A052799 (PARI) a(n)={if(n>=4, 2*4!*binomial(n,4)*abs(stirling(n-4,2,1)), 0)} \\ _Andrew Howroyd_, Aug 08 2020
%Y A052799 Cf. A052766.
%K A052799 easy,nonn
%O A052799 0,7
%A A052799 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052799 New name using e.g.f., _Vaclav Kotesovec_, Oct 01 2013