A368091 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p} r, where P(n, k) are the partitions of n with length k.

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 7, 2, 1, 0, 5, 10, 7, 2, 1, 0, 6, 22, 18, 7, 2, 1, 0, 7, 28, 34, 18, 7, 2, 1, 0, 8, 50, 62, 50, 18, 7, 2, 1, 0, 9, 60, 121, 86, 50, 18, 7, 2, 1, 0, 10, 95, 182, 189, 118, 50, 18, 7, 2, 1

Offset: 0

Author

Peter Luschny, Dec 11 2023

Keywords

• nonn
• tabl

Examples

			Table T(n, k) starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 2,  1]
  [3] [0, 3,  2,   1]
  [4] [0, 4,  7,   2,  1]
  [5] [0, 5, 10,   7,  2,  1]
  [6] [0, 6, 22,  18,  7,  2,  1]
  [7] [0, 7, 28,  34, 18,  7,  2, 1]
  [8] [0, 8, 50,  62, 50, 18,  7, 2, 1]
  [9] [0, 9, 60, 121, 86, 50, 18, 7, 2, 1]
		

Crossrefs

Cf. A368090, A074141, A023855, A006906 (row sums).

Programs

SageMath

def T(n, k):
    return sum(product(r 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)])