A052765 Expansion of e.g.f.: -x^2*(log(1-x))^3.

0, 0, 0, 0, 0, 120, 1080, 8820, 75600, 701568, 7091280, 77961840, 928778400, 11937347136, 164802429792, 2433765035520, 38299272560640, 639999894048768, 11320441140625920, 211340086405770240, 4153253573744179200, 85710868976433254400, 1853386172505676892160

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..448
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 722

Crossrefs

Cf. A052754, A052758.

Programs

Maple

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

Mathematica

CoefficientList[Series[-x^2*(Log[1-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

PARI

x='x+O('x^30); concat(vector(5), Vec(serlaplace(x^2*log(-1/(-1+x))^3))) \\ G. C. Greubel, Sep 05 2018

PARI

a(n)={if(n>=2, 3!*n*(n-1)*abs(stirling(n-2,3,1)), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

E.g.f.: x^2*log(-1/(-1+x))^3.

Recurrence: a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=120, (16*n-48*n^2-4*n^3-n^6+13*n^4+48)*a(n) + (n^2+61*n-26*n^3+3*n^4+3*n^5-42)*a(n+1) + (-9*n+15*n^2-3*n^3-3*n^4)*a(n+2) + (n^3-n)*a(n+3) = 0.

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

a(n) = n*A052758(n-1) = 3!*n*(n-1)*abs(Stirling1(n-2,3)) for n >= 2. - Andrew Howroyd, Aug 08 2020

Extensions

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