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.
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, 0, 8, 52, 122, 142, 136, 96, 128, 0, 9, 86, 197, 325, 284, 272, 192, 256, 0, 10, 100, 350, 502, 650, 568, 544, 384, 512
Offset: 0
Examples
Triangle T(n, k) starts: [0] [1] [1] [0, 2] [2] [0, 3, 4] [3] [0, 4, 6, 8] [4] [0, 5, 17, 12, 16] [5] [0, 6, 22, 34, 24, 32] [6] [0, 7, 43, 71, 68, 48, 64] [7] [0, 8, 52, 122, 142, 136, 96, 128] [8] [0, 9, 86, 197, 325, 284, 272, 192, 256] [9] [0, 10, 100, 350, 502, 650, 568, 544, 384, 512]
Programs
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SageMath
def T(n, k): return sum(product(r+1 for r in p) for p in Partitions(n, length=k)) for n in range(10): print([T(n, k) for k in range(n + 1)])