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%I #10 Dec 15 2023 06:27:32
%S 1,0,2,0,3,4,0,4,6,8,0,5,17,12,16,0,6,22,34,24,32,0,7,43,71,68,48,64,
%T 0,8,52,122,142,136,96,128,0,9,86,197,325,284,272,192,256,0,10,100,
%U 350,502,650,568,544,384,512
%N Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p}(r + 1), where P(n, k) are the partitions of n with length k.
%e Triangle T(n, k) starts:
%e [0] [1]
%e [1] [0, 2]
%e [2] [0, 3, 4]
%e [3] [0, 4, 6, 8]
%e [4] [0, 5, 17, 12, 16]
%e [5] [0, 6, 22, 34, 24, 32]
%e [6] [0, 7, 43, 71, 68, 48, 64]
%e [7] [0, 8, 52, 122, 142, 136, 96, 128]
%e [8] [0, 9, 86, 197, 325, 284, 272, 192, 256]
%e [9] [0, 10, 100, 350, 502, 650, 568, 544, 384, 512]
%o (SageMath)
%o def T(n, k):
%o return sum(product(r+1 for r in p) for p in Partitions(n, length=k))
%o for n in range(10): print([T(n, k) for k in range(n + 1)])
%Y Cf. A238963, A368091, A074141 (row sums).
%K nonn,tabl
%O 0,3
%A _Peter Luschny_, Dec 11 2023
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