A359107 Triangle read by rows, T(n, k) = Sum_{j=0..k} Stirling2(n, j) = Sum_{j=0..k} A048993(n, j).

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 8, 14, 15, 0, 1, 16, 41, 51, 52, 0, 1, 32, 122, 187, 202, 203, 0, 1, 64, 365, 715, 855, 876, 877, 0, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140, 0, 1, 256, 3281, 11051, 18002, 20648, 21110, 21146, 21147

Offset: 0

Author

Peter Luschny, Dec 27 2022

Keywords

• nonn
• tabl

Comments

T(n, k) is the number of partitions of an n-set that contain at most k nonempty subsets.

Examples

			Triangle T(n, k) starts:
[0] [1]
[1] [0, 1]
[2] [0, 1,   2]
[3] [0, 1,   4,    5]
[4] [0, 1,   8,   14,    15]
[5] [0, 1,  16,   41,    51,    52]
[6] [0, 1,  32,  122,   187,   202,   203]
[7] [0, 1,  64,  365,   715,   855,   876,   877]
[8] [0, 1, 128, 1094,  2795,  3845,  4111,  4139,  4140]
[9] [0, 1, 256, 3281, 11051, 18002, 20648, 21110, 21146, 21147]
		

Crossrefs

Cf. A048993, A000110 (T(n, n)), A359355 (T(2*n, n)), A359109 (row sums).

Without column k=0 the same as A102661.

Programs

Maple

with(ListTools): ps := L -> PartialSums(L):
Flatten([seq(ps([seq(Stirling2(n, k), k = 0..n)]), n = 0..10)]);