A052758 -id:A052758 - cp's OEIS frontend

A052770 Expansion of e.g.f. log(-1/(-1+x))^4*x.

0, 0, 0, 0, 0, 120, 1440, 14280, 141120, 1462104, 16148160, 191051520, 2421936000, 32840519712, 475109002368, 7312951114560, 119419875993600, 2063254247160192, 37617587514390528, 721991172931734528

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 727

Cf. A052758, A052783.

spec := [S,{B=Cycle(Z),S=Prod(Z,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative
A052770 := proc(n)
        egf := log(-1/(-1+x))^4*x ;
        coeftayl(%,x=0,n)*n! ;
end proc:
seq(A052770(n),n=0..20) ; # R. J. Mathar, Jan 13 2025

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (n^8+6*n^7-46*n-21*n^2-5*n^4-34*n^5+24+74*n^3+n^6)*a(n) + (-4*n^7+49*n^3-70*n^2+25*n^4-54*n^5-30*n^6+24*n)*a(n+1) + (115*n^4+48*n^5+6*n^6+19*n^2+12*n+80*n^3)*a(n+2) + (-16*n-70*n^3-60*n^2-30*n^4-4*n^5)*a(n+3) + (n^4+6*n^3+11*n^2+6*n)*a(n+4), a(5)=120}.

A052745 Expansion of e.g.f. log(-1/(-1+x))^2*x.

Original entry on oeis.org

0, 0, 0, 6, 24, 110, 600, 3836, 28224, 235224, 2191680, 22584672, 255087360, 3134139840, 41620400640, 594082771200, 9070900715520, 147531542054400, 2546434166169600, 46489412442009600, 895079522340864000, 18125736166340812800, 385129713617510400000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 701

Cf. A052758, A052770, A052783.

[0] cat [(-1)^(n+1)*2*n*StirlingFirst(n-1, 2): n in [1..30]]; // Vincenzo Librandi, Jul 08 2015

spec := [S,{B=Cycle(Z),S=Prod(Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative
A052745 := proc(n)
    (log(1/(1-x)))^2*x ;
    coeftayl(%,x=0,n)*n! ;
end proc:
seq(A052745(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

Range[0, 30]! CoefficientList[Series[Log[-1/(-1 + x)]^2 x,{x, 0, 30}], x] (* Vincenzo Librandi, Jul 08 2015 *)

makelist((-1)^(n+1)*2*n*stirling1(n-1, 2), n, 0, 20); /* Bruno Berselli, May 25 2011 */

Recurrence: a(1)=0, a(2)=0, a(3)=6, (-n+n^4+n^3-3*n^2+2)*a(n)+(-2*n^3-3*n^2+2*n)*a(n+1)+(n^2+n)*a(n+2)=0.

a(n) = (-1)^(n+1)*2*n*Stirling1(n-1, 2). - Vladeta Jovovic, Nov 08 2003

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)}.

A052765 Expansion of e.g.f.: -x^2*(log(1-x))^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 1080, 8820, 75600, 701568, 7091280, 77961840, 928778400, 11937347136, 164802429792, 2433765035520, 38299272560640, 639999894048768, 11320441140625920, 211340086405770240, 4153253573744179200, 85710868976433254400, 1853386172505676892160

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..448
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 722

Cf. A052754, A052758.

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

CoefficientList[Series[-x^2*(Log[1-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

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

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

E.g.f.: x^2*log(-1/(-1+x))^3.
Recurrence: a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=120, (16*n-48*n^2-4*n^3-n^6+13*n^4+48)*a(n) + (n^2+61*n-26*n^3+3*n^4+3*n^5-42)*a(n+1) + (-9*n+15*n^2-3*n^3-3*n^4)*a(n+2) + (n^3-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
a(n) = n*A052758(n-1) = 3!*n*(n-1)*abs(Stirling1(n-2,3)) for n >= 2. - Andrew Howroyd, Aug 08 2020

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

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A052770 Expansion of e.g.f. log(-1/(-1+x))^4*x.

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A052745 Expansion of e.g.f. log(-1/(-1+x))^2*x.

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

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A052765 Expansion of e.g.f.: -x^2*(log(1-x))^3.

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