A077012 -id:A077012 - cp's OEIS frontend

A078921 Signed variant of A077012.

1, -1, 2, 2, -3, 6, -6, 8, -12, 24, 24, -30, 40, -60, 120, -120, 144, -180, 240, -360, 720, 720, -840, 1008, -1260, 1680, -2520, 5040, -5040, 5760, -6720, 8064, -10080, 13440, -20160, 40320, 40320, -45360, 51840, -60480, 72576, -90720, 120960, -181440, 362880, -362880, 403200, -453600, 518400, -604800, 725760, -907200, 1209600, -1814400, 3628800

Offset: 1

Wouter Meeussen, Dec 14 2002

Row sums give A024167.

			Triangle starts:
     1
    -1,    2
     2,   -3,    6
    -6,    8,  -12,    24
    24,  -30,   40,   -60,  120
  -120,  144, -180,   240, -360,   720
   720, -840, 1008, -1260, 1680, -2520, 5040
  ...

Cf. A000254, A024167, A077012, A078922.

Table[Table[ -(-1)^(n-k+1) n/(n-k+1), {k, 1, n}] (n-1)!, {n, 1, 12}]

T(n, k) = -(-1)^(n-k+1)*(n/(n-k+1))*(n-1)!.
E.g.f.: log(1+x)/(1-y*x). - Vladeta Jovovic, Feb 07 2003
Sum_{n>=1} Sum_{k=1..n} 1/T(n, k) = (e^2+1)/(4*e). - Amiram Eldar, Jun 29 2025

A166350 Triangle read by rows: T(n,m) = m!, n >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 6, 24, 1, 2, 6, 24, 120, 1, 2, 6, 24, 120, 720, 1, 2, 6, 24, 120, 720, 5040, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 1, 2, 6, 24, 120, 720, 5040, 40320

Offset: 1

Paul Curtz, Oct 12 2009

			Triangle begins:
  1;
  1, 2;
  1, 2, 6;
  1, 2, 6, 24;
  1, 2, 6, 24, 120;
  1, 2, 6, 24, 120, 720;
  1, 2, 6, 24, 120, 720, 5040;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
Index entries for sequences related to factorial numbers

Cf. A000142, A002260, A094310, A077012, A079210.
Cf. A014454.
Row sums give A007489.

import Data.List (inits)
a166350 n k = a166350_tabl !! (n-1) !! (n-1)
a166350_row n = a166350_tabl !! (n-1)
a166350_tabl = tail $ inits $ tail a000142_list
-- Reinhard Zumkeller, Nov 11 2013

Flatten[Table[Range[n]!,{n,11}]] (* Harvey P. Dale, Jan 06 2012 *)
Module[{nn=20,fs},fs=Range[nn]!;Table[Take[fs,n],{n,nn}]]//Flatten (* Harvey P. Dale, Jun 14 2020 *)

T(n,m) = A000142(m).

Definition clarified - R. J. Mathar, Oct 14 2009

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

A335442 List enumerated in lexicographic order of (n, s, k), where for each n >= 1, for each s a subset of 1..n with n-1 elements, and for each k in 0..n-1, we give the value of (Sum_{t subset of s, Card(t)=k} Product_{x in t} x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 1, 5, 6, 1, 6, 11, 6, 1, 7, 14, 8, 1, 8, 19, 12, 1, 9, 26, 24, 1, 10, 35, 50, 24, 1, 11, 41, 61, 30, 1, 12, 49, 78, 40, 1, 13, 59, 107, 60, 1, 14, 71, 154, 120, 1, 15, 85, 225, 274, 120, 1, 16, 95, 260, 324, 144, 1, 17, 107, 307, 396, 180, 1, 18, 121, 372, 508, 240, 1, 19, 137, 461, 702, 360, 1, 20, 155, 580, 1044, 720

Offset: 1

Luc Rousseau, Jun 10 2020

This sequence can be viewed as a triangle made of square blocks of increasing sizes: 1 X 1, 2 X 2, and so on. In block number n >= 1, the bottom-right corner is n!, the top-right one (n-1)!. The first row of block number n, which is row number binomial(n, 2) if we number the rows according to their second value, is the list of the unsigned Stirling numbers of the first kind, reversed. The subsequence {last element of row n} is A077012.
Block number n crops up for instance when studying -log(1-r), where r = e^(i*2*Pi/n); this is Sum_{K>=1} r^K/K = Sum_{K>=0} Sum_{L=1..n} r^L/(nK+L);
the term for K in the first sum, if put in the same denominator and if no simplification is carried out, has a numerator which is a combination of all (nK)^I * r^J; their coefficients are precisely the elements of block number n.

			Table begins:
  +---+
  | 1 |
  +---+----+
  | 1    1 |
  | 1    2 |
  +--------+----+
  | 1    3    2 |
  | 1    4    3 |
  | 1    5    6 |
  +-------------+----+
  | 1    6   11    6 |
  | 1    7   14    8 |
  | 1    8   19   12 |
  | 1    9   26   24 |
  +------------------+----+
  | 1   10   35   50   24 |
  | 1   11   41   61   30 |
  | 1   12   49   78   40 |
  | 1   13   59  107   60 |
  | 1   14   71  154  120 |
  +-----------------------+----+
  | 1   15   85  225  274  120 |
  | 1   16   95  260  324  144 |
  | 1   17  107  307  396  180 |
  | 1   18  121  372  508  240 |
  | 1   19  137  461  702  360 |
  | 1   20  155  580 1044  720 |
  +----------------------------+----+
  | 1   21  175  735 1624 1764  720 |
  | 1   22  190  820 1849 2038  840 |
  | 1   23  207  925 2144 2412 1008 |
  | 1   24  226 1056 2545 2952 1260 |
  | 1   25  247 1219 3112 3796 1680 |
  | 1   26  270 1420 3929 5274 2520 |
  | 1   27  295 1665 5104 8028 5040 |
  +---------------------------------+----+
  etc.

Luc Rousseau, Program for the computation of A335442 (SWI-Prolog)

Cf. A000142, A008275, A077012.

A165969 Triangle read by rows: T(n,m) = A094310(n,m)*A120070(n+1,m), 1 <= m <= n.

Original entry on oeis.org

3, 16, 5, 90, 36, 14, 576, 252, 128, 54, 4200, 1920, 1080, 600, 264, 34560, 16200, 9600, 5940, 3456, 1560, 317520, 151200, 92400, 60480, 39312, 23520, 10800, 3225600, 1552320, 967680, 655200, 451584, 302400, 184320, 85680, 35925120, 17418240, 11007360, 7620480, 5443200, 3870720, 2643840, 1632960, 766080

Offset: 1

Paul Curtz, Oct 02 2009

The second array mentioned in the comment in A129326.

			Triangle begins
        3;
       16,       5;
       90,      36,     14;
      576,     252,    128,     54;
     4200,    1920,   1080,    600,    264;
    34560,   16200,   9600,   5940,   3456,   1560;
   317520,  151200,  92400,  60480,  39312,  23520,  10800;
  3225600, 1552320, 967680, 655200, 451584, 302400, 184320, 85680;

Cf. A077012, A079210.

A120070 := proc(n, m) n^2-m^2 ; end proc:
A094310 := proc(n,m) n!/m ; end proc:
A165969 := proc(n,m) A094310(n,m)*A120070(n+1,m) ; end proc: # R. J. Mathar, Jan 23 2011

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A078921 Signed variant of A077012.

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A166350 Triangle read by rows: T(n,m) = m!, n >= 1.

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

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A335442 List enumerated in lexicographic order of (n, s, k), where for each n >= 1, for each s a subset of 1..n with n-1 elements, and for each k in 0..n-1, we give the value of (Sum_{t subset of s, Card(t)=k} Product_{x in t} x).

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A165969 Triangle read by rows: T(n,m) = A094310(n,m)*A120070(n+1,m), 1 <= m <= n.

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