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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
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
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 11 2023
STATUS
approved