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

Author

Peter Luschny, Dec 11 2023

Keywords

• nonn
• tabl

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]
		

Crossrefs

Cf. A238963, A368091, A074141 (row sums).

Programs

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)])