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A368090 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.

%I A368090 #10 Dec 15 2023 06:27:32
%S A368090 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 A368090 0,8,52,122,142,136,96,128,0,9,86,197,325,284,272,192,256,0,10,100,
%U A368090 350,502,650,568,544,384,512
%N A368090 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 A368090 Triangle T(n, k) starts:
%e A368090   [0] [1]
%e A368090   [1] [0,  2]
%e A368090   [2] [0,  3,   4]
%e A368090   [3] [0,  4,   6,   8]
%e A368090   [4] [0,  5,  17,  12,  16]
%e A368090   [5] [0,  6,  22,  34,  24,  32]
%e A368090   [6] [0,  7,  43,  71,  68,  48,  64]
%e A368090   [7] [0,  8,  52, 122, 142, 136,  96, 128]
%e A368090   [8] [0,  9,  86, 197, 325, 284, 272, 192, 256]
%e A368090   [9] [0, 10, 100, 350, 502, 650, 568, 544, 384, 512]
%o A368090 (SageMath)
%o A368090 def T(n, k):
%o A368090     return sum(product(r+1 for r in p) for p in Partitions(n, length=k))
%o A368090 for n in range(10): print([T(n, k) for k in range(n + 1)])
%Y A368090 Cf. A238963, A368091, A074141 (row sums).
%K A368090 nonn,tabl
%O A368090 0,3
%A A368090 _Peter Luschny_, Dec 11 2023