A052758 Expansion of e.g.f.: -(log(1-x))^3*x.

0, 0, 0, 0, 24, 180, 1260, 9450, 77952, 709128, 7087440, 77398200, 918257472, 11771602128, 162251002368, 2393704535040, 37647052591104, 628913396701440, 11123162442408960, 207662678687208960

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 714

Programs

Maple

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative
A052758 := proc(n)
    (log(1/(1-x)))^3*x ;
    coeftayl(%,x=0,n)*n! ;
end proc:
seq(A052758(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

Mathematica

CoefficientList[Series[-(Log[1-x])^3*x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)
Join[{0,0,0,0}, RecurrenceTable[{a[4] == 24, a[5] == 180, a[6] == 1260, (4*n^4 -n^6 -7*n -9*n^2 -3*n^5 +6 +10*n^3)*a[n] + (3*n^5 +12*n^4 +4*n^3 -13*n^2 +6*n)*a[n+1] +(-12*n^3 -3*n^4 -9*n^2)*a[n+2] == -(n^3 +3*n^2 + 2*n)*a[n+3]}, 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))^3*x ))) \\ G. C. Greubel, Sep 05 2018

Formula

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

Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (4*n^4-n^6-7*n-9*n^2-3*n^5+6+10*n^3)*a(n) + (3*n^5+12*n^4+4*n^3-13*n^2+6*n)*a(n+1) + (-12*n^3-3*n^4-9*n^2)*a(n+2) + (n^3+3*n^2+2*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

Extensions

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