A362221 Irregular triangle read by rows: T(n, k) is the number of partitions of n into exactly k distinct parts between the members of [k^2].
1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 1, 0, 0, 5, 1, 0, 0, 7, 2, 0, 0, 7, 3, 0, 0, 8, 5, 0, 0, 8, 6, 1, 0, 0, 8, 9, 1, 0, 0, 7, 11, 2, 0, 0, 7, 15, 3, 0, 0, 5, 18, 5, 0, 0, 4, 23, 7, 0, 0, 3, 27, 10, 1, 0, 0, 2, 34, 13, 1, 0, 0, 1, 38, 18, 2
Offset: 1
Examples
The irregular triangle begins:
1;
0;
0, 1;
0, 1;
0, 2;
0, 1, 1;
0, 1, 1;
0, 0, 2;
0, 0, 3;
0, 0, 4, 1;
0, 0, 5, 1;
0, 0, 7, 2;
0, 0, 7, 3;
0, 0, 8, 5;
0, 0, 8, 6, 1;
...
T(11,3) = 5 since we have: 1+2+8, 1+3+7, 1+4+6, 2+3+6, 2+4+5.
Programs
-
Mathematica
Flatten[Table[Length[Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@# &&Length[#]==k&]], {n, 23}, {k, Floor[(Sqrt[8n+1]-1)/2]}]]