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