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%I #25 Aug 09 2020 00:32:16
%S 0,0,0,0,0,120,720,4620,33600,276192,2540160,25874640,289301760,
%T 3523208832,46425899520,658169366400,9988896153600,161590513766400,
%U 2775695618949120,50455787382604800,967644983144448000
%N Expansion of e.g.f.: (log(1-x))^2*x^3.
%C Previous name was: A simple grammar.
%H G. C. Greubel, <a href="/A052766/b052766.txt">Table of n, a(n) for n = 0..449</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=723">Encyclopedia of Combinatorial Structures 723</a>
%F E.g.f.: log(-1/(-1+x))^2*x^3.
%F Recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, (n^4-7*n^2-3*n^3+15*n+18)*a(n) + (8*n-2*n^3+5*n^2-20)*a(n+1) + (-3*n+n^2+2)*a(n+2) = 0.
%F a(n) ~ 2*(n-1)! * (log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Sep 30 2013
%F a(n) = n*A052754(n-1) = 2*n*(n-1)*(n-2)*abs(Stirling1(n-3,2)) for n >= 3. - _Andrew Howroyd_, Aug 08 2020
%p spec := [S,{B=Cycle(Z),S=Prod(Z,Z,Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t CoefficientList[Series[(Log[1-x])^2*x^3, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%t Join[{0,0,0,0,0}, RecurrenceTable[{a[5] == 120, a[6] == 720, (n^4 -7*n^2 -3*n^3 +15*n +18)*a[n] + (8*n -2*n^3 +5*n^2 -20)*a[n+1] == -(-3*n +n^2 + 2)*a[n+2]}, a, {n, 5, 30}]] (* _G. C. Greubel_, Sep 05 2018 *)
%o (PARI) x='x+O('x^30); concat(vector(5), Vec(serlaplace(log(-1/(-1+x))^2*x^3))) \\ _G. C. Greubel_, Sep 05 2018
%o (PARI) a(n)={if(n>=3, 2*n*(n-1)*(n-2)*abs(stirling(n-3,2,1)), 0)} \\ _Andrew Howroyd_, Aug 08 2020
%Y Cf. A052754, A052759.
%K easy,nonn
%O 0,6
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name, using e.g.f., by _Vaclav Kotesovec_, Sep 30 2013
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