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A339030 T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.

%I A339030 #19 Nov 28 2020 12:46:47
%S A339030 1,0,1,0,2,1,0,3,6,1,0,4,24,8,1,0,5,85,50,10,1,0,6,300,280,75,12,1,0,
%T A339030 7,1071,1540,525,105,14,1,0,8,3976,8456,3570,840,140,16,1,0,9,15219,
%U A339030 47208,24381,6552,1260,180,18,1
%N A339030 T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.
%e A339030 Triangle starts:
%e A339030 0: [1]
%e A339030 1: [0, 1]
%e A339030 2: [0, 2, 1]
%e A339030 3: [0, 3, 6,     1]
%e A339030 4: [0, 4, 24,    8,     1]
%e A339030 5: [0, 5, 85,    50,    10,    1]
%e A339030 6: [0, 6, 300,   280,   75,    12,   1]
%e A339030 7: [0, 7, 1071,  1540,  525,   105,  14,   1]
%e A339030 8: [0, 8, 3976,  8456,  3570,  840,  140,  16,  1]
%e A339030 9: [0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1]
%e A339030 .
%e A339030 T(4,0) = 0  = 0*card({})
%e A339030 T(4,1) = 4  = 4*card({1|2|3|4}).
%e A339030 T(4,2) = 24 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
%e A339030             + 2*card({12|34, 13|24, 14|23}).
%e A339030 T(4,3) = 8  = 2*card({123|4, 124|3, 134|2, 1|234}).
%e A339030 T(4,4) = 1  = 1*card({1234}).
%e A339030 .
%e A339030 Seen as the projection of a 2-dimensional statistic this is, for n = 6:
%e A339030 [  0   0    0     0     0    0   0]
%e A339030 [  0   0    0     0     0    0   6]
%e A339030 [  0   0    0    45   180   75   0]
%e A339030 [  0   0   20   180    80    0   0]
%e A339030 [  0   0   30    45     0    0   0]
%e A339030 [  0   0   12     0     0    0   0]
%e A339030 [  0   1    0     0     0    0   0]
%e A339030 The row sum projection gives row 6 of this triangle, and the column sum projection gives [0, 1, 62, 270, 260, 75, 6], which appears in a decapitated version as row 5 in A321331.
%o A339030 (SageMath)
%o A339030 def A339030Row(n):
%o A339030     if n == 0: return [1]
%o A339030     M = matrix(n + 1)
%o A339030     for k in (1..n):
%o A339030         for p in SetPartitions(n):
%o A339030             if p.max_block_size() == k:
%o A339030                 M[k, len(p)] += p.cardinality()
%o A339030     return [sum(M[k, j] for j in (0..n)) for k in (0..n)]
%o A339030 for n in (0..9): print(A339030Row(n))
%Y A339030 Cf. A005493 with 1 prepended are the row sums.
%Y A339030 Cf. A080510, A262071, A321331.
%K A339030 nonn,tabl
%O A339030 0,5
%A A339030 _Peter Luschny_, Nov 22 2020