A362208 Irregular triangle read by rows: T(n, k) is the number of compositions (ordered partitions) of n into exactly k distinct parts between the members of [k^2].

1, 0, 0, 2, 0, 2, 0, 4, 0, 2, 6, 0, 2, 6, 0, 0, 12, 0, 0, 18, 0, 0, 24, 24, 0, 0, 30, 24, 0, 0, 42, 48, 0, 0, 42, 72, 0, 0, 48, 120, 0, 0, 48, 144, 120, 0, 0, 48, 216, 120, 0, 0, 42, 264, 240, 0, 0, 42, 360, 360, 0, 0, 30, 432, 600, 0, 0, 24, 552, 840, 0, 0, 18, 648, 1200, 720

Offset: 1

Author

Stefano Spezia, Apr 11 2023

Keywords

• nonn
• tabf

Examples

			The irregular triangle begins:
    1;
    0;
    0, 2;
    0, 2;
    0, 4;
    0, 2,  6;
    0, 2,  6;
    0, 0, 12;
    0, 0, 18;
    0, 0, 24,  24;
    0, 0, 30,  24;
    0, 0, 42,  48;
    0, 0, 42,  72;
    0, 0, 48, 120;
    0, 0, 48, 144, 120;
    ...
T(7,3) = 6 since we have: 1+2+4, 1+4+2, 2+1+4, 2+4+1, 4+1+2, 4+2+1.
		

Crossrefs

Cf. A000290, A003056 (row lengths), A072574, A216652.

Cf. A362209, A362221 (unordered partitions).

Programs

Mathematica

Flatten[Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n,All,Range[k^2]], UnsameQ@@#&], Length[#]==k&]], {n, 21}, {k, Floor[(Sqrt[8n+1]-1)/2]}]] (* After Gus Wiseman in A072574 *)