A373029 - OEIS

%I #13 May 20 2024 08:56:59

%S 1,0,1,0,1,1,0,2,1,1,0,2,1,1,1,0,3,1,1,1,1,0,4,2,2,1,1,1,0,5,3,1,2,1,

%T 1,1,0,6,3,1,2,2,1,1,1,0,8,4,3,2,2,2,1,1,1,0,10,5,3,2,3,2,2,1,1,1,0,

%U 12,6,4,2,3,3,2,2,1,1,1,0,15,7,6,3,3,4,3,2,2,1,1,1,0,18,9,6,4,3,4,4,3,2,2,1,1,1

%N Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of distinct partitions p of n such that max(p) is a multiple of k.

%F For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i) * Product_{j=1..k*i-1} (1+x^j).

%e Triangle begins:

%e   1;

%e   0,  1;

%e   0,  1, 1;

%e   0,  2, 1, 1;

%e   0,  2, 1, 1, 1;

%e   0,  3, 1, 1, 1, 1;

%e   0,  4, 2, 2, 1, 1, 1;

%e   0,  5, 3, 1, 2, 1, 1, 1;

%e   0,  6, 3, 1, 2, 2, 1, 1, 1;

%e   0,  8, 4, 3, 2, 2, 2, 1, 1, 1;

%e   0, 10, 5, 3, 2, 3, 2, 2, 1, 1, 1;

%e   0, 12, 6, 4, 2, 3, 3, 2, 2, 1, 1, 1;

%e   0, 15, 7, 6, 3, 3, 4, 3, 2, 2, 1, 1, 1;

%e   0, 18, 9, 6, 4, 3, 4, 4, 3, 2, 2, 1, 1, 1;

%Y Row sums give A373030.

%Y Column k=0..3 give A000007, A000009, A026838, A372893.

%Y T(2n,n) gives A000009.

%Y Cf. A363048.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, May 20 2024