A052753 Expansion of e.g.f.: log(1-x)^4.

0, 0, 0, 0, 24, 240, 2040, 17640, 162456, 1614816, 17368320, 201828000, 2526193824, 33936357312, 487530074304, 7463742249600, 121367896891776, 2089865973021696, 37999535417459712, 727710096185266176, 14642785817771802624, 308902349883623731200, 6818239581643475251200

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 709

Crossrefs

Column k=4 of A225479.

Cf. A000454, A052517.

Programs

Maple

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

Mathematica

CoefficientList[Series[(Log[1-x])^4, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

PARI

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

PARI

a(n) = {4!*stirling(n,4,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

Formula

E.g.f.: log(-1/(-1+x))^4.

Recurrence: {a(1)=0, a(0)=0, a(2)=0, (1+4*n+6*n^2+4*n^3+n^4)*a(n+1) + (-4*n^3-15-18*n^2-28*n)*a(n+2) + (6*n^2+24*n+25)*a(n+3) + (-4*n-10)*a(n+4)+a(n+5), a(3)=0, a(4)=24}.

a(n) ~ (n-1)! * 2*log(n)*(2*log(n)^2 + 6*gamma*log(n) - Pi^2 + 6*gamma^2), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013

a(n) = 24*A000454(n) = 4!*(-1)^n*Stirling1(n,4). - Andrew Howroyd, Jul 27 2020

Extensions

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