A368091 - OEIS

%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