A052745 -id:A052745 - cp's OEIS frontend

A052754 Expansion of e.g.f.: (log(1-x))^2*x^2.

0, 0, 0, 0, 24, 120, 660, 4200, 30688, 254016, 2352240, 24108480, 271016064, 3316135680, 43877957760, 624306009600, 9505324339200, 154205312163840, 2655567756979200, 48382249157222400, 929788248840192000, 18796669969158144000, 398766195659497881600

Offset: 0

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

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..449
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 710

Cf. A052745, A052747.

I:=[24,120]; [0,0,0,0] cat [n le 2 select I[n] else (n*(n+3)*(2*n-1)*Self(n-1) - (n-1)^2*(n+2)*(n+3)*Self(n-2))/(n*(n+1)): n in [1..30]]; // G. C. Greubel, Sep 05 2018

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

CoefficientList[Series[(Log[1-x])^2*x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)
Join[{0,0,0,0}, RecurrenceTable[{a[4] == 24, a[5] == 120, (n^4 - 6*n^2 - n^3 + 4*n + 8)*a[n] + (7*n - 2*n^3 + n^2 - 6)*a[n + 1] == -(n^2 - n)*a[n + 2]}, a, {n, 4, 30}]] (* G. C. Greubel, Sep 05 2018 *)

x='x+O('x^30); concat(vector(4), Vec(serlaplace(log(-1/(-1+x))^2* x^2))) \\ G. C. Greubel, Sep 05 2018

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

E.g.f.: log(-1/(-1+x))^2*x^2.
Recurrence: a(1)=0, a(2)=0, a(3)=0, a(4)=24, (n^4-6*n^2-n^3+4*n+8)*a(n) + (7*n-2*n^3+n^2-6)*a(n+1) + (n^2-n)*a(n+2) = 0.
a(n) ~ (n-1)! * 2*(log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 01 2013
a(n) = n*A052745(n-1) = 2*n*(n-1)*abs(Stirling1(n-2,2)) for n >= 2. - Andrew Howroyd, Aug 08 2020

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

A052783 Expansion of e.g.f. x*log(-1/(-1+x))^5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 12600, 168000, 2116800, 26938800, 355509000, 4920379200, 71753338800, 1104107484480, 17923866760800, 306665482905600, 5521899225024000, 104470579944195840, 2073203785324575360

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 740

Cf. A052745, A052758, A052770.

spec := [S,{B=Cycle(Z),S=Prod(Z,B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative
A052783 := proc(n)
    (log(1/(1-x)))^5*x ;
    coeftayl(%,x=0,n)*n! ;
end proc:
seq(A052783(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (500*n^3 + 55*n^2 - 300*n^4 - n^10 + 120 - 224*n^5 - 20*n^8 + 146*n^6 - 10*n^9 + 60*n^7 - 326*n)*a(n) + (225*n^7 - n^4 - 299*n^5 + 120*n + 60*n^8 + 5*n^9 + 195*n^6 - 446*n^2 + 501*n^3)*a(n + 1) + ( - 535*n^3 - 895*n^4 - 10*n^8 - 130*n^7 - 300*n^2 - 1135*n^5 - 595*n^6)*a(n + 2) + (130*n^6 + 10*n^7 + 200*n + 1425*n^3 + 1330*n^4 + 615*n^5 + 790*n^2)*a(n + 3) + ( - 150*n - 455*n^2 - 60*n^5 - 5*n^6 - 510*n^3 - 260*n^4)*a(n + 4) + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a(n + 5)}.

cp's OEIS Frontend

A052754 Expansion of e.g.f.: (log(1-x))^2*x^2.

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