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A052758 Expansion of e.g.f.: -(log(1-x))^3*x.

%I A052758 #24 Jan 20 2025 06:36:36
%S A052758 0,0,0,0,24,180,1260,9450,77952,709128,7087440,77398200,918257472,
%T A052758 11771602128,162251002368,2393704535040,37647052591104,
%U A052758 628913396701440,11123162442408960,207662678687208960
%N A052758 Expansion of e.g.f.: -(log(1-x))^3*x.
%C A052758 Previous name was: A simple grammar.
%H A052758 G. C. Greubel, <a href="/A052758/b052758.txt">Table of n, a(n) for n = 0..448</a>
%H A052758 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=714">Encyclopedia of Combinatorial Structures 714</a>
%F A052758 E.g.f.: log(-1/(-1+x))^3*x.
%F A052758 Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (4*n^4-n^6-7*n-9*n^2-3*n^5+6+10*n^3)*a(n) + (3*n^5+12*n^4+4*n^3-13*n^2+6*n)*a(n+1) + (-12*n^3-3*n^4-9*n^2)*a(n+2) + (n^3+3*n^2+2*n)*a(n+3) = 0}.
%F A052758 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
%p A052758 spec := [S,{B=Cycle(Z),S=Prod(B,B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052758 # alternative
%p A052758 A052758 := proc(n)
%p A052758     (log(1/(1-x)))^3*x ;
%p A052758     coeftayl(%,x=0,n)*n! ;
%p A052758 end proc:
%p A052758 seq(A052758(n),n=0..20) ; # _R. J. Mathar_, Jan 20 2025
%t A052758 CoefficientList[Series[-(Log[1-x])^3*x, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 01 2013 *)
%t A052758 Join[{0,0,0,0}, RecurrenceTable[{a[4] == 24, a[5] == 180, a[6] == 1260, (4*n^4 -n^6 -7*n -9*n^2 -3*n^5 +6 +10*n^3)*a[n] + (3*n^5 +12*n^4 +4*n^3 -13*n^2 +6*n)*a[n+1] +(-12*n^3 -3*n^4 -9*n^2)*a[n+2] == -(n^3 +3*n^2 + 2*n)*a[n+3]}, a, {n, 4, 30}]] (* _G. C. Greubel_, Sep 05 2018 *)
%o A052758 (PARI) x='x+O('x^30); concat(vector(4), Vec(serlaplace(log(-1/(-1+x))^3*x ))) \\ _G. C. Greubel_, Sep 05 2018
%K A052758 easy,nonn
%O A052758 0,5
%A A052758 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052758 New name using e.g.f., _Vaclav Kotesovec_, Oct 01 2013