A052759 -id:A052759 - cp's OEIS frontend

A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

4, 36, 9, 576, 144, 64, 14400, 3600, 1600, 900, 518400, 129600, 57600, 32400, 20736, 25401600, 6350400, 2822400, 1587600, 1016064, 705600, 1625702400, 406425600, 180633600, 101606400, 65028096, 45158400, 33177600, 131681894400

Offset: 1

Johannes W. Meijer, Jul 21 2009

The hypergeometric function 3F2([1,n+1,n+1],[n+2,n+2],z) = (n+1)^2*Li2(z)/z^(n+1) - MN(z;n)/(n!^2*z^n) for n >= 1, with Li2(z) the dilogarithm. The polynomial coefficients of MN(z;n) lead to the triangle given above.
We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.
The generating function for the EG1[3,n] coefficients of the EG1 matrix, see A162005, is GFEG1(z;m=2) = 1/(1-z)*(3*zeta(3)/2-2*z*log(2)* 3F2([1,1,1],[2,2],z) + sum((2^(1-2*n)* factorial(2*n-1)*z^(n+1)*3F2([1,n+1,n+1],[n+2,n+2],z))/(factorial(n+1)^2), n=1..infinity)).
The zeros of the MN(z;n) polynomials for larger values of n get ever closer to the unit circle and resemble the full moon, hence we propose to call the MN(z;n) the moon polynomials.

			The first few rows of the triangle are:
  [4]
  [36, 9]
  [576, 144, 64]
  [14400, 3600, 1600, 900]
The first few MN(z;n) polynomials are:
  MN(z;n=1) = 4
  MN(z;n=2) = 36 + 9*z
  MN(z;n=3) = 576 + 144*z + 64*z^2
  MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3

Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.

Paolo Xausa, Table of n, a(n) for n = 1..5050 (rows 1..100 of the triangle, flattened).
Eric Weisstein's World of Mathematics, Dilogarithm.

A162995 is a scaled version of this triangle.
A001819(n)*(n+1)^2 equals the row sums for n>=1.
A162991 and A162992 equal the first and second right hand columns.
A001048, A052747, A052759, A052778, A052794 are related to the square root of the first five right hand columns.
A001044, A162993 and A162994 equal the first, second and third left hand columns.
A000142, A001710, A002301, A133799, A129923, A001715 are related to the square root of the first six left hand columns.
A027451(n+1) equals the denominators of M(z, n)/(n!)^2.
A129202(n)/A129203(n) = (n+1)^2*Li2(z=1)/(Pi^2) = (n+1)^2/6.
Cf. A002378 and A035287.

a := proc(n, m): ((n+1)!/m)^2 end: seq(seq(a(n, m), m=1..n), n=1..7); # Johannes W. Meijer, revised Nov 29 2012

Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* Paolo Xausa, Mar 30 2024 *)

a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.

A052778 E.g.f.: x^4*log(-1/(-1+x)).

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 360, 1680, 10080, 72576, 604800, 5702400, 59875200, 691891200, 8717829120, 118879488000, 1743565824000, 27360571392000, 457312407552000, 8109673360588800, 152056375511040000, 3005349539512320000, 62444484876533760000, 1360632459941314560000

Offset: 0

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

Previous name was: A simple grammar.

Amiram Eldar, Table of n, a(n) for n = 0..450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 735.

Cf. A052747, A052794, A052759.

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

a[n_] := If[n < 5, 0, n!/(n - 4)]; Array[a, 20, 0] (* Amiram Eldar, Oct 07 2020 *)
With[{nn=30},CoefficientList[Series[x^4 Log[-1/(x-1)],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 28 2021 *)

E.g.f.: x^4*log(-1/(-1+x)).
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, (-n^2+3*n+4)*a(n) + (n-3)*a(n+1)}.
a(n) = n! / (n-4) (n > 4). - Olivier Gérard, Jun 13 2001
Sum_{n>=5} 1/a(n) = 49/6 - 3*e. - Amiram Eldar, Oct 07 2020
Sum_{n>=5} (-1)^(n+1)/a(n) = 5/e - 11/6. - Amiram Eldar, Aug 20 2022

New name using e.g.f. from Vaclav Kotesovec, Oct 07 2020

A052794 E.g.f.: -x^5*log(1-x).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 2520, 13440, 90720, 725760, 6652800, 68428800, 778377600, 9686476800, 130767436800, 1902071808000, 29640619008000, 492490285056000, 8688935743488000, 162193467211776000, 3193183885731840000, 66117689869271040000, 1436223152160276480000

Offset: 0

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

Amiram Eldar, Table of n, a(n) for n = 0..450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 751.

Cf. A052747, A052759, A052778.

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

a[n_] := If[n < 6, 0, n!/(n - 5)]; Array[a, 20, 0] (* Amiram Eldar, Oct 07 2020 *)

E.g.f.: x^5*log(-1/(-1+x)).
Recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (-n^2+4*n+5)*a(n) + (-4+n)*a(n+1)=0.
From Amiram Eldar, Oct 07 2020: (Start)
a(n) = n! / (n-5) for n > 5.
Sum_{n>=6} 1/a(n) = 87/8 - 4*e. (End)
Sum_{n>=6} (-1)^n/a(n) = 53/24 - 6/e. - Amiram Eldar, Aug 20 2022

A052766 Expansion of e.g.f.: (log(1-x))^2*x^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 720, 4620, 33600, 276192, 2540160, 25874640, 289301760, 3523208832, 46425899520, 658169366400, 9988896153600, 161590513766400, 2775695618949120, 50455787382604800, 967644983144448000

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 723

Cf. A052754, A052759.

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

CoefficientList[Series[(Log[1-x])^2*x^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
Join[{0,0,0,0,0}, RecurrenceTable[{a[5] == 120, a[6] == 720, (n^4 -7*n^2 -3*n^3 +15*n +18)*a[n] + (8*n -2*n^3 +5*n^2 -20)*a[n+1] == -(-3*n +n^2 + 2)*a[n+2]}, a, {n, 5, 30}]] (* G. C. Greubel, Sep 05 2018 *)

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

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

E.g.f.: log(-1/(-1+x))^2*x^3.
Recurrence: a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=120, (n^4-7*n^2-3*n^3+15*n+18)*a(n) + (8*n-2*n^3+5*n^2-20)*a(n+1) + (-3*n+n^2+2)*a(n+2) = 0.
a(n) ~ 2*(n-1)! * (log(n) + gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013
a(n) = n*A052754(n-1) = 2*n*(n-1)*(n-2)*abs(Stirling1(n-3,2)) for n >= 3. - Andrew Howroyd, Aug 08 2020

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

A058298 Triangle n!/(n-k), 1 <= k < n, read by rows.

Original entry on oeis.org

2, 3, 6, 8, 12, 24, 30, 40, 60, 120, 144, 180, 240, 360, 720, 840, 1008, 1260, 1680, 2520, 5040, 5760, 6720, 8064, 10080, 13440, 20160, 40320, 45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880, 403200, 453600, 518400, 604800, 725760, 907200, 1209600, 1814400, 3628800

Offset: 2

Leroy Quet, Dec 07 2000

Together with 1, numbers n such that n divides k! if and only if k! >= n. - Charles R Greathouse IV, Aug 16 2016

			Triangle begins:
      2;
      3,     6;
      8,    12,    24;
     30,    40,    60,   120;
    144,   180,   240,   360,   720;
    840,  1008,  1260,  1680,  2520,   5040;
   5760,  6720,  8064, 10080, 13440,  20160,  40320;
  45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
Florian Unger and Jonathan Krebs, MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs, arXiv:2309.02938 [math.ST], 2023.

Columns k=1..5 are A001048(n-1), A052747, A052759, A052778, A052794.
Row sums are A052881.
Cf. A019739. A077012.

Flatten[Table[n!/(n-k),{n,2,10},{k,n-1}]] (* Harvey P. Dale, Jul 23 2014 *)

PARI
```
T(n,k)={if(kAndrew Howroyd, Aug 08 2020
```

Sum_{n>=2} Sum_{k=1..n-1} 1/T(n, k) = e/2 (A019739). - Amiram Eldar, Jun 29 2025

cp's OEIS Frontend

A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

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A052778 E.g.f.: x^4*log(-1/(-1+x)).

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A052794 E.g.f.: -x^5*log(1-x).

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A052766 Expansion of e.g.f.: (log(1-x))^2*x^3.

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A058298 Triangle n!/(n-k), 1 <= k < n, read by rows.

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