%I #7 Dec 14 2023 08:57:22
%S 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,
%T 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,
%U 189,118,50,18,7,2,1
%N 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.
%e Table T(n, k) starts:
%e [0] [1]
%e [1] [0, 1]
%e [2] [0, 2, 1]
%e [3] [0, 3, 2, 1]
%e [4] [0, 4, 7, 2, 1]
%e [5] [0, 5, 10, 7, 2, 1]
%e [6] [0, 6, 22, 18, 7, 2, 1]
%e [7] [0, 7, 28, 34, 18, 7, 2, 1]
%e [8] [0, 8, 50, 62, 50, 18, 7, 2, 1]
%e [9] [0, 9, 60, 121, 86, 50, 18, 7, 2, 1]
%o (SageMath)
%o def T(n, k):
%o return sum(product(r for r in p) for p in Partitions(n, length=k))
%o for n in range(10): print([T(n, k) for k in range(n + 1)])
%Y Cf. A368090, A074141, A023855, A006906 (row sums).
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Dec 11 2023
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