A052766 Expansion of e.g.f.: (log(1-x))^2*x^3.

0, 0, 0, 0, 0, 120, 720, 4620, 33600, 276192, 2540160, 25874640, 289301760, 3523208832, 46425899520, 658169366400, 9988896153600, 161590513766400, 2775695618949120, 50455787382604800, 967644983144448000

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 723

Crossrefs

Cf. A052754, A052759.

Programs

Maple

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

Mathematica

CoefficientList[Series[(Log[1-x])^2*x^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
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 *)

PARI

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

PARI

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

Formula

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

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.

a(n) ~ 2*(n-1)! * (log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013

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

Extensions

New name, using e.g.f., by Vaclav Kotesovec, Sep 30 2013