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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.

%I A368091 #7 Dec 14 2023 08:57:22
%S A368091 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 A368091 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 A368091 189,118,50,18,7,2,1
%N 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.
%e A368091 Table T(n, k) starts:
%e A368091   [0] [1]
%e A368091   [1] [0, 1]
%e A368091   [2] [0, 2,  1]
%e A368091   [3] [0, 3,  2,   1]
%e A368091   [4] [0, 4,  7,   2,  1]
%e A368091   [5] [0, 5, 10,   7,  2,  1]
%e A368091   [6] [0, 6, 22,  18,  7,  2,  1]
%e A368091   [7] [0, 7, 28,  34, 18,  7,  2, 1]
%e A368091   [8] [0, 8, 50,  62, 50, 18,  7, 2, 1]
%e A368091   [9] [0, 9, 60, 121, 86, 50, 18, 7, 2, 1]
%o A368091 (SageMath)
%o A368091 def T(n, k):
%o A368091     return sum(product(r for r in p) for p in Partitions(n, length=k))
%o A368091 for n in range(10): print([T(n, k) for k in range(n + 1)])
%Y A368091 Cf. A368090, A074141, A023855, A006906 (row sums).
%K A368091 nonn,tabl
%O A368091 0,5
%A A368091 _Peter Luschny_, Dec 11 2023