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A052753 Expansion of e.g.f.: log(1-x)^4.

%I A052753 #24 Jul 27 2020 16:45:32
%S A052753 0,0,0,0,24,240,2040,17640,162456,1614816,17368320,201828000,
%T A052753 2526193824,33936357312,487530074304,7463742249600,121367896891776,
%U A052753 2089865973021696,37999535417459712,727710096185266176,14642785817771802624,308902349883623731200,6818239581643475251200
%N A052753 Expansion of e.g.f.: log(1-x)^4.
%C A052753 Previous name was: A simple grammar.
%H A052753 G. C. Greubel, <a href="/A052753/b052753.txt">Table of n, a(n) for n = 0..448</a>
%H A052753 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=709">Encyclopedia of Combinatorial Structures 709</a>
%F A052753 E.g.f.: log(-1/(-1+x))^4.
%F A052753 Recurrence: {a(1)=0, a(0)=0, a(2)=0, (1+4*n+6*n^2+4*n^3+n^4)*a(n+1) + (-4*n^3-15-18*n^2-28*n)*a(n+2) + (6*n^2+24*n+25)*a(n+3) + (-4*n-10)*a(n+4)+a(n+5), a(3)=0, a(4)=24}.
%F A052753 a(n) ~ (n-1)! * 2*log(n)*(2*log(n)^2 + 6*gamma*log(n) - Pi^2 + 6*gamma^2), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Sep 30 2013
%F A052753 a(n) = 24*A000454(n) = 4!*(-1)^n*Stirling1(n,4). - _Andrew Howroyd_, Jul 27 2020
%p A052753 spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052753 CoefficientList[Series[(Log[1-x])^4, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o A052753 (PARI) x='x+O('x^30); concat(vector(4), Vec(serlaplace((log(1-x))^4))) \\ _G. C. Greubel_, Aug 30 2018
%o A052753 (PARI) a(n) = {4!*stirling(n,4,1)*(-1)^n} \\ _Andrew Howroyd_, Jul 27 2020
%Y A052753 Column k=4 of A225479.
%Y A052753 Cf. A000454, A052517.
%K A052753 easy,nonn
%O A052753 0,5
%A A052753 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052753 New name using e.g.f., _Vaclav Kotesovec_, Sep 30 2013