A052790 Expansion of e.g.f.: x^2*log(1-x)^4.

0, 0, 0, 0, 0, 0, 720, 10080, 114240, 1270080, 14621040, 177629760, 2292618240, 31485168000, 459767275968, 7126635035520, 117007217832960, 2030137891891200, 37138576448883456, 714734162773420032, 14439823458634690560, 305638240397811793920, 6764967047810572812288

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Original name: a simple grammar.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 747

Crossrefs

Cf. A052754, A052770.

Programs

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[x^2 Log[-1/(x-1)]^4,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2016 *)

PARI

a(n)={if(n>=2, 4!*n*(n-1)*abs(stirling(n-2,4,1)), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

E.g.f.: x^2*log(-1/(-1+x))^4.

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (32*n-464*n^2-21*n^6-22*n^5+48*n^3+n^8+2*n^7+384+160*n^4)*a(n) + (105*n^4-360-14*n^6-121*n^2-4*n^7+642*n-296*n^3+48*n^5)*a(n+1) + (-84*n+24*n^5+179*n^2+6*n^6-35*n^4-90*n^3)*a(n+2) + (14*n^2+12*n^3-8*n-14*n^4-4*n^5)*a(n+3) + (-n^2+n^4-2*n+2*n^3)*a(n+4)}.

a(n) = n*A052770(n-1) = 4!*n*(n-1)*abs(Stirling1(n-2,4)) for n >= 2. - Andrew Howroyd, Aug 08 2020

Extensions

Name changed and terms a(20) and beyond from Andrew Howroyd, Aug 08 2020