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A339031 T(n, k) = k*P(n, k), where P(n, k) is the number of partitions of an n-set with k blocks, the largest of which has the size n - k + 1. Triangle T(n, k) for 0 <= k <= n, read by rows.

%I A339031 #12 Nov 28 2020 12:46:42
%S A339031 1,0,1,0,1,2,0,1,6,3,0,1,8,18,4,0,1,10,30,40,5,0,1,12,45,80,75,6,0,1,
%T A339031 14,63,140,175,126,7,0,1,16,84,224,350,336,196,8,0,1,18,108,336,630,
%U A339031 756,588,288,9
%N A339031 T(n, k) = k*P(n, k), where P(n, k) is the number of partitions of an n-set with k blocks, the largest of which has the size n - k + 1. Triangle T(n, k) for 0 <= k <= n, read by rows.
%F A339031 T(n, k) = k*binomial(n, k - 1) for n >= 1 and 0 < k < n, and T(n, 0) = 0^n, T(n, n) = n.
%e A339031 Triangle starts:
%e A339031 0: [1]
%e A339031 1: [0, 1]
%e A339031 2: [0, 1,  2]
%e A339031 3: [0, 1,  6,   3]
%e A339031 4: [0, 1,  8,  18,   4]
%e A339031 5: [0, 1, 10,  30,  40,   5]
%e A339031 6: [0, 1, 12,  45,  80,  75,   6]
%e A339031 7: [0, 1, 14,  63, 140, 175, 126,   7]
%e A339031 8: [0, 1, 16,  84, 224, 350, 336, 196,   8]
%e A339031 9: [0, 1, 18, 108, 336, 630, 756, 588, 288, 9]
%e A339031 .
%e A339031 T(4, 1) =  1 = 1*card({1234})
%e A339031 T(4, 2) =  8 = 2*card({123|4, 124|3, 134|2, 1|234})
%e A339031 T(4, 3) = 18 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
%e A339031 T(4, 4) =  4 = 4*card({1|2|3|4})
%p A339031 A339031 := proc(n, k) if k = 0 then 0^n elif k = n then n
%p A339031 else k*binomial(n, k-1) fi end:
%p A339031 seq(seq(A339031(n, k), k=0..n), n=0..9);
%o A339031 (SageMath) # Shows the combinatorial interpretation.
%o A339031 def A339031Row(n):
%o A339031     if n == 0: return [1]
%o A339031     M = matrix(n + 2)
%o A339031     for k in (1..n):
%o A339031         for p in SetPartitions(n):
%o A339031             if p.max_block_size() == k:
%o A339031                 M[len(p), k] += p.cardinality()
%o A339031     return [M[k, n-k+1] for k in (0..n)]
%o A339031 for n in (0..9): print(A339031Row(n))
%Y A339031 Cf. A339032 (row sums), A339030.
%K A339031 nonn,tabl
%O A339031 0,6
%A A339031 _Peter Luschny_, Nov 22 2020