A052754 Expansion of e.g.f.: (log(1-x))^2*x^2.

0, 0, 0, 0, 24, 120, 660, 4200, 30688, 254016, 2352240, 24108480, 271016064, 3316135680, 43877957760, 624306009600, 9505324339200, 154205312163840, 2655567756979200, 48382249157222400, 929788248840192000, 18796669969158144000, 398766195659497881600

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..449
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 710

Crossrefs

Cf. A052745, A052747.

Programs

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

Maple

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

Mathematica

CoefficientList[Series[(Log[1-x])^2*x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)
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 *)

PARI

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

PARI

a(n)={if(n>=2, 2*n*(n-1)*abs(stirling(n-2,2,1)), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

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

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.

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

a(n) = n*A052745(n-1) = 2*n*(n-1)*abs(Stirling1(n-2,2)) for n >= 2. - Andrew Howroyd, Aug 08 2020

Extensions

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013