A130191 -id:A130191 - cp's OEIS frontend

A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

1, 2, 2, 5, 12, 6, 15, 64, 72, 24, 52, 350, 660, 480, 120, 203, 2024, 5670, 6720, 3600, 720, 877, 12460, 48552, 83160, 71400, 30240, 5040, 4140, 81638, 424536, 983808, 1201200, 806400, 282240, 40320

Offset: 1

Tilman Piesk, Dec 07 2013

T(n,k) is the number of preferential arrangements with k levels of partitions of the set {1...n}.
2*T(n,k) is the number of formulas in first order logic that have an n-place predicate and k runs of A's and E's (universal and existential quantifiers, compare runs of 0's ans 1's counted by A005811), but don't include a negator.
4*T(n,k) is the number of such formulas that may include an negator.
T(n,k) is the number of partitions of an n-set into colored blocks, such that exactly k colors are used. T(3,2) = 12: 1a|23b, 1b|23a, 13a|2b, 13b|2a, 12a|3b, 12b|3a, 1a|2a|3b, 1b|2b|3a, 1a|2b|3a, 1b|2a|3b, 1a|2b|3b, 1b|2a|3a. - Alois P. Heinz, Sep 01 2019

			Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
Compare descriptions of A083355 and A232598.
a(3,1)=5:
{1,2,3}
{1,2}{3}
{1,3}{2}
{2,3}{1}
{1}{2}{3}
a(3,2)=12:
{1,2}:{3}   {3}:{1,2}
{1,3}:{2}   {2}:{1,3}
{2,3}:{1}   {1}:{2,3}
{1}{2}:{3}   {3}:{1}{2}
{1}{3}:{2}   {2}:{1}{3}
{2}{3}:{1}   {1}:{2}{3}
a(3,3)=6:
{1}:{2}:{3}
{1}:{3}:{2}
{2}:{1}:{3}
{2}:{3}:{1}
{3}:{1}:{2}
{3}:{2}:{1}
Triangle begins:
       k = 1     2      3      4       5      6      7     8           sums
1          1                                                              1
2          2     2                                                        4
3          5    12      6                                                23
4         15    64     72     24                                        175
5         52   350    660    480     120                               1662
6        203  2024   5670   6720    3600    720                       18937
7        877 12460  48552  83160   71400  30240   5040               251729
8       4140 81638 424536 983808 1201200 806400 282240 40320        3824282

Tilman Piesk, First 100 rows, flattened
Tilman Piesk, Preferential arrangements of set partitions (Wikiversity)

A008277 (Stirling2), A039810 (square of Stirling2), A000110 (Bell), A000142 (factorials), A083355 (row sums: number of preferential arrangements), A232598 (number of preferential arrangements by number of blocks).

Cf. A130191.

S2 = A008277 (Stirling numbers of the second kind).
(S2)^2 = A039810 (matrix square of S2).
T(n,k) = ((S2)^2)(n,k) * k! = Sum(k<=i<=n) [ S2(n,i) * S2(i,k) ] * k!.
T(n,1) = Bell(n) = A000110(n).
T(n,2) = A052896(n).
T(n,n) = n! = A000142(n).
T(n,n-1) = n!*(n-1) = A062119(n).

A383206 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * Stirling2(j,k).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 11, 9, 1, 0, 49, 71, 18, 1, 0, 257, 575, 245, 30, 1, 0, 1539, 4957, 3120, 625, 45, 1, 0, 10299, 45829, 39697, 11480, 1330, 63, 1, 0, 75905, 454015, 517790, 201677, 33250, 2506, 84, 1, 0, 609441, 4804191, 6999785, 3513762, 770007, 81774, 4326, 108, 1

Offset: 0

Seiichi Manyama, Apr 19 2025

			Triangle starts:
  1;
  0,     1;
  0,     3,     1;
  0,    11,     9,     1;
  0,    49,    71,    18,     1;
  0,   257,   575,   245,    30,    1;
  0,  1539,  4957,  3120,   625,   45,  1;
  0, 10299, 45829, 39697, 11480, 1330, 63, 1;
  ...

Columns k=0..3 give A000007, A004211 (for n > 0), A383207, A383208.
Row sums give A380228.
Cf. A130191.

T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 2));

E.g.f. of column k (with leading zeros): (exp(f(x)) - 1)^k / k! with f(x) = (exp(2*x) - 1)/2.

cp's OEIS Frontend

A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

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