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

0,3

Links

Table of n, a(n) for n=0..54.

Example

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]

Prog

(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)])

Crossrefs

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

Sequence in context: A099091 A227595 A078436 * A209705 A181289 A229032

Adjacent sequences: A368087 A368088 A368089 * A368091 A368092 A368093

Keyword

nonn,tabl

Author

Peter Luschny, Dec 11 2023

Status

approved