A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

1, 3, 2, 13, 12, 6, 75, 74, 60, 24, 541, 540, 510, 360, 120, 4683, 4682, 4620, 4080, 2520, 720, 47293, 47292, 47166, 45360, 36960, 20160, 5040, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320, 7087261, 7087260, 7086750, 7068600, 6882120, 6048000, 4142880, 1814400, 362880

Offset: 1

Author

N. J. A. Sloane, Jun 24 2003

Keywords

• nonn
• tabl
• easy

Comments

Interpolates between A000670 and factorials.

From Thomas Scheuerle, Apr 25 2022: (Start)

Number of preferential arrangements of n labeled elements when at least k ranks are required.

This sequence starts for k and n with offset 1. If it would start with k = 0, we would observe in column k = 0 an exact copy of column k = 1 with a preceding one at n = 0, k = 0. The difference between 0 ranks and one rank (all in the same rank) is only for n = 0 where k = 0 allows zero-filled ranks as an valid arrangement, too. (End)

Examples

			Triangle begins with T(n,k):
   k=   1,   2,   3,   4,   5
  n=1   1
  n=2   3,   2
  n=3  13,  12,   6
  n=4  75,  74,  60,  24
  n=5 541, 540, 510, 360, 120
...
From _Thomas Scheuerle_, Apr 25 2022: (Start)
If we would add n = 0, k = 0 to the data of this sequence:
   k=   0,   1,   2,
  n=0   1
  n=1   1,   1
  n=2   3,   3,   2
...
T(n, 3) with 3 preceding zeros is: 0,0,0,6,60,510,4620,...
This sequence has the e.g.f.: (e^x-1)^3/(2-e^x).
.
13 arrangements for n = 3 and k = 1 (one rank required):
1,2,3  1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
12 arrangements for n = 3 and k = 2 (two ranks required):
1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
6 arrangements for n = 3 and k = 3 (three ranks required):
1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
. (End)
		

Crossrefs

Mirror image of array in A084417.

Cf. A005649, A069321 (row sums).

A000670(n) (column k = 1), A052875(n) (column k = 2), A102232(n) (column k = 3).

Cf. A000142, A008277.

Programs

Maple

T := (n,k)->sum(i!*Stirling2(n,i),i=k..n): seq(seq(T(n,k),k=1..n),n=1..10);

PARI

row(n) = vector(n, k, sum(i=k, n, i!*stirling(n, i, 2))); \\ Michel Marcus, Apr 20 2022

Formula

E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - Vladeta Jovovic, Sep 14 2003

T(n, k) = Sum_{m = k..n} A090582(n + 1, m + 1).

From Thomas Scheuerle, Apr 25 2022: (Start)

Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 is not part of data.

Sum_{k = 1..n} T(n, k) = A069321(n).

T(n, 0) = A000670(n). Column k = 0 is not part of data.

T(n, 1) = A000670(n), for n > 0.

T(n, 2) = A052875(n).

T(n, 3) = A102232(n).

T(n, n) = n! = A000142. (End)

Extensions

More terms from Emeric Deutsch, May 11 2004

More terms from Michel Marcus, Apr 20 2022