A052747 -id:A052747 - 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.

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

Original entry on oeis.org

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

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

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

A092824 Farey-factorial numerators.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 30, 40, 48, 60, 72, 80, 90, 96, 120, 144, 180, 240, 288, 360, 432, 480, 540, 576, 600, 720, 840, 1008, 1260, 1440, 1680, 2016, 2160, 2520, 2880, 3024, 3360, 3600, 3780, 4032, 4200, 4320, 5040, 5760, 6720, 8064, 10080

Offset: 1

Clark Kimberling, Mar 06 2004

nonn

The last number in the n-th segment is n!. Let f(n) be the first number in segment n; except for initial terms, f is A001048 and A059171. Let g(n) be the second number in segment n; except for initial terms, g is A052747. Except for the initial terms, the number of numbers in segment n is given by A015614.

			The sequence begins with these segments:
  1
  2
  3 4 6
  8 12 16 18 24
For the next segment, start with these Farey fractions of order 5:
  1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/5.
Multiply these by 5! to get
  30 40 48 60 72 80 90 96 120.

Cf. A000142, A001048, A015614, A059171, A052747.

f[n_] := n! * Table[a/b, {b, 1, n}, {a, 1, b}] // Flatten // Union // Rest; Flatten[Table[f[n], {n, 1, 8}] /. {} -> {1}][[1 ;; 51]] (* Jean-François Alcover, May 18 2011 *)

Let S(n) be the set of integers an!/b, where a/b ranges through the positive Farey fractions of order n. A092824 is the increasing sequence of integers in the union of the sets S(n), for n>=1.

A153805 Decimal expansion of 3-e.

Original entry on oeis.org

2, 8, 1, 7, 1, 8, 1, 7, 1, 5, 4, 0, 9, 5, 4, 7, 6, 4, 6, 3, 9, 7, 1, 2, 5, 2, 8, 6, 4, 7, 3, 3, 7, 5, 0, 2, 2, 4, 2, 7, 5, 2, 9, 0, 6, 3, 0, 0, 0, 4, 0, 4, 2, 5, 0, 3, 3, 0, 3, 2, 3, 7, 2, 2, 7, 5, 9, 2, 3, 3, 6, 9, 6, 4, 6, 4, 5, 2, 4, 0, 5, 4, 2, 8, 6, 1, 7, 8, 2, 1, 4, 7, 4, 8, 3, 3, 5, 7, 2, 5, 7, 2, 5, 3

Offset: 0

Omar E. Pol, Jan 28 2009

			0.2817181715409547646397125286473375...

G. C. Greubel, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A001113, A052747, A098916. Essentially the same as A121239.

RealDigits[3 - E, 10, 150][[1]] (* Harvey P. Dale, Nov 29 2011 *)

From Amiram Eldar, Aug 06 2020: (Start)
Equals Sum_{k>=0} k/(k + 2)!.
Equals Sum_{k>=0} 1/(k! * (k + 2) * (k + 3)) = Sum_{k>=3} 1/A052747(k).
Equals Sum_{k>=2} 1/(k! * k * (k - 1)) = Sum_{k>=4} 1/A098916(k). (End)
Area between log(x) and log(x)^2, i.e., Integral_{x=1..exp(1)} (log(x) - log(x)^2) dx. - Martin Renner, May 17 2022

A162992 The second right hand column of triangle A162990.

Original entry on oeis.org

36, 144, 1600, 32400, 1016064, 45158400, 2687385600, 205752960000, 19670999040000, 2294425328025600, 320461058211840000, 52778155948277760000, 10118415696592896000000, 2233485391313240064000000

Offset: 2

Johannes W. Meijer, Jul 21 2009

A052747(n+1) equals the square root of a(n).

A162991 is the first right hand column of triangle A162990.

Array[((#+1)!/(#-1))^2 &, 20, 2] (* Paolo Xausa, Apr 01 2024 *)

a(n) = ((n+1)!/(n-1))^2 for n = 2, 3, ... .

a(n) = A052747(n+1)^2.

A271376 Integers n such that n!/(n-2) + 1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 12, 13, 21, 24, 37, 64, 68, 136, 169, 216, 276, 435, 617, 753, 1722, 1775, 2762, 3974, 5006, 5931, 7480, 11442, 11896, 13200, 13534, 23240, 27971, 31867

Offset: 1

Altug Alkan, Apr 05 2016

nonn

Corresponding primes are 7, 13, 41, 181, 1009, 453601, ...

			3 is a term because 2*3 + 1 = 7 is prime.
4 is a term because 1*3*4 + 1 = 13 is prime.
5 is a term because 1*2*4*5 + 1 = 41 is prime.
6 is a term because 1*2*3*5*6 + 1 = 181 is prime.
7 is a term because 1*2*3*4*6*7 + 1 = 1009 is prime.

Cf. A002981, A052747.

[n: n in [3..500] | IsPrime(Factorial(n)  div (n-2) + 1)]; // Vincenzo Librandi, Apr 07 2016

Select[Range[3, 2000], PrimeQ[( #! / (# - 2) + 1)] &] (* Vincenzo Librandi, Apr 07 2016 *)

lista(nn) = for(n=3, nn, if(ispseudoprime(n!/(n-2)+1), print1(n, ", ")));

ABC2 $a!/($a-2) + 1
a: from 3 to 100000

a(23) from Charles R Greathouse IV, Apr 05 2016
a(24)-a(27) from Charles R Greathouse IV, Apr 06 2016
a(28)-a(32) from Charles R Greathouse IV, Apr 18 2016
a(33) from Charles R Greathouse IV, Apr 30 2016
a(34) from Charles R Greathouse IV, May 09 2016

A291322 Numbers k such that k!/(k-2) - 1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 18, 21, 23, 26, 46, 70, 83, 156, 553, 591, 741, 790, 1430, 2139, 5509, 11429, 11881, 14343

Offset: 1

Amiram Eldar and Altug Alkan, Aug 22 2017

Corresponding primes are 5, 11, 179, 6719, 51839, 453599, ...
Initial terms common with A271376 (that generate twin primes): 3, 4, 6, 10, 21.
Union of this sequence and A271376 is the sequence {3, 4, 5, 6, 7, 8, 9, 10, 12, 13, ...}.

			3 is a term because 2*3 - 1 = 5 is prime.
4 is a term because 1*3*4 - 1 = 11 is prime.
6 is a term because 1*2*3*5*6 - 1 = 179 is prime.

Cf. A002982, A052747, A271376.

[n: n in [3..1000] | IsPrime(Factorial(n)  div (n-2) - 1)];

Select[Range[3, 6000], PrimeQ[( #! / (# - 2) - 1)] &]

lista(nn) = for(n=3, nn, if(ispseudoprime(n!/(n-2)-1), print1(n, ", ")));

PFGW
```
ABC2 $a!/($a-2) - 1
a: from 3 to 6000
```

a(22)-a(23) from Giovanni Resta, Aug 23 2017

a(24) from Michael S. Branicky, Dec 31 2024

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

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

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PARI

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A092824 Farey-factorial numerators.

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Mathematica

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A153805 Decimal expansion of 3-e.

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