A052754 - OEIS

%I #27 Sep 08 2022 08:44:59

%S 0,0,0,0,24,120,660,4200,30688,254016,2352240,24108480,271016064,

%T 3316135680,43877957760,624306009600,9505324339200,154205312163840,

%U 2655567756979200,48382249157222400,929788248840192000,18796669969158144000,398766195659497881600

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

%C Previous name was: A simple grammar.

%H G. C. Greubel, <a href="/A052754/b052754.txt">Table of n, a(n) for n = 0..449</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=710">Encyclopedia of Combinatorial Structures 710</a>

%F E.g.f.: log(-1/(-1+x))^2*x^2.

%F Recurrence: a(1)=0, a(2)=0, a(3)=0, a(4)=24, (n^4-6*n^2-n^3+4*n+8)*a(n) + (7*n-2*n^3+n^2-6)*a(n+1) + (n^2-n)*a(n+2) = 0.

%F a(n) ~ (n-1)! * 2*(log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 01 2013

%F a(n) = n*A052745(n-1) = 2*n*(n-1)*abs(Stirling1(n-2,2)) for n >= 2. - _Andrew Howroyd_, Aug 08 2020

%p spec := [S,{B=Cycle(Z),S=Prod(B,B,Z,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t CoefficientList[Series[(Log[1-x])^2*x^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 01 2013 *)

%t Join[{0,0,0,0}, RecurrenceTable[{a[4] == 24, a[5] == 120, (n^4 - 6*n^2 - n^3 + 4*n + 8)*a[n] + (7*n - 2*n^3 + n^2 - 6)*a[n + 1] == -(n^2 - n)*a[n + 2]}, a, {n, 4, 30}]] (* _G. C. Greubel_, Sep 05 2018 *)

%o (PARI) x='x+O('x^30); concat(vector(4), Vec(serlaplace(log(-1/(-1+x))^2* x^2))) \\ _G. C. Greubel_, Sep 05 2018

%o (PARI) a(n)={if(n>=2, 2*n*(n-1)*abs(stirling(n-2,2,1)), 0)} \\ _Andrew Howroyd_, Aug 08 2020

%o (Magma) I:=[24,120]; [0,0,0,0] cat [n le 2 select I[n] else (n*(n+3)*(2*n-1)*Self(n-1) - (n-1)^2*(n+2)*(n+3)*Self(n-2))/(n*(n+1)): n in [1..30]]; // _G. C. Greubel_, Sep 05 2018

%Y Cf. A052745, A052747.

%K easy,nonn

%O 0,5

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E New name using e.g.f., _Vaclav Kotesovec_, Oct 01 2013