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%I #9 Jun 13 2023 07:55:41
%S 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,
%T 203,0,1,64,365,715,855,876,877,0,1,128,1094,2795,3845,4111,4139,4140,
%U 0,1,256,3281,11051,18002,20648,21110,21146,21147
%N Triangle read by rows, T(n, k) = Sum_{j=0..k} Stirling2(n, j) = Sum_{j=0..k} A048993(n, j).
%C T(n, k) is the number of partitions of an n-set that contain at most k nonempty subsets.
%e Triangle T(n, k) starts:
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, 1, 2]
%e [3] [0, 1, 4, 5]
%e [4] [0, 1, 8, 14, 15]
%e [5] [0, 1, 16, 41, 51, 52]
%e [6] [0, 1, 32, 122, 187, 202, 203]
%e [7] [0, 1, 64, 365, 715, 855, 876, 877]
%e [8] [0, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140]
%e [9] [0, 1, 256, 3281, 11051, 18002, 20648, 21110, 21146, 21147]
%p with(ListTools): ps := L -> PartialSums(L):
%p Flatten([seq(ps([seq(Stirling2(n, k), k = 0..n)]), n = 0..10)]);
%Y Cf. A048993, A000110 (T(n, n)), A359355 (T(2*n, n)), A359109 (row sums).
%Y Without column k=0 the same as A102661.
%K nonn,tabl
%O 0,6
%A _Peter Luschny_, Dec 27 2022
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