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

1,2

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)

Links

Table of n, a(n) for n=1..45.

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)

Example

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)

Maple

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

Prog

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

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.

Sequence in context: A252669 A131050 A219374 * A210604 A005352 A095131

Adjacent sequences: A084413 A084414 A084415 * A084417 A084418 A084419

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jun 24 2003

Extensions

More terms from Emeric Deutsch, May 11 2004

More terms from Michel Marcus, Apr 20 2022

Status

approved