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

0,5

Links

Table of n, a(n) for n=0..65.

Example

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]

Prog

(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)])

Crossrefs

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

Sequence in context: A131103 A180653 A259100 * A365004 A096652 A322133

Adjacent sequences: A368088 A368089 A368090 * A368092 A368093 A368094

Keyword

nonn,tabl

Author

Peter Luschny, Dec 11 2023

Status

approved