A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

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

Offset

1,2

Comments

The sum of the rows gives A000254 (Stirling numbers of first kind). The first column and the leading diagonal are factorials given by A000142 with offsets of 0 and 1.

T(n,k) is the number of length k cycles in all permutations of {1..n}.

Second diagonal gives A001048(n). - Anton Zakharov, Oct 24 2016

T(n,k) is the number of permutations of [n] with all elements of [k] in a single cycle. To prove this result, let m denote the length of the cycle containing {1,..,k}. Letting m run from k to n, we obtain T(n,k) = Sum_{m=k..n} (C(n-k,m-k)*(m-1)!*(n-m)!) = n!/k. See an example below. - Dennis P. Walsh, May 24 2020

Links

Alois P. Heinz, Rows n = 1..141, flattened

Formula

E.g.f. for column k: x^k/(k*(1-x)).

T(n,k)*k = n*n! = A001563(n).

Example

Triangle begins as:
      1;
      2,     1;
      6,     3,     2;
     24,    12,     8,     6;
    120,    60,    40,    30,   24;
    720,   360,   240,   180,  144,  120;
   5040,  2520,  1680,  1260, 1008,  840,  720;
  40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040;
  ...
T(4,2) counts the 12 permutations of [4] with elements 1 and 2 in the same cycle, namely, (1 2)(3 4), (1 2)(3)(4), (1 2 3)(4), (1 3 2)(4), (1 2 4)(3), (1 4 2)(3), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), and (1 4 3 2). - Dennis P. Walsh, May 24 2020

Maple

seq(seq(n!/k, k=1..n), n=1..10);

Mathematica

Table[n!/k, {n, 10}, {k, n}]//Flatten
Table[n!/Range[n], {n, 10}]//Flatten (* Harvey P. Dale, Mar 12 2016 *)

Crossrefs

Cf. A000142, A000254, A001563, A001710, A002301.

Cf. A061579, A094307, A110468, A133799.

Cf. A129825. - Johannes W. Meijer, Jun 18 2009

Sequence in context: A125901 A094307 A097905 * A165908 A121281 A232467

Adjacent sequences: A094307 A094308 A094309 * A094311 A094312 A094313

Keyword

nonn,tabl

Author

Amarnath Murthy, Apr 29 2004

Extensions

More terms from Philippe Deléham, Jun 11 2005

Status

approved