A362221 - OEIS

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].

%I #9 Apr 16 2023 20:37:57

%S 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,

%T 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,

%U 4,23,7,0,0,3,27,10,1,0,0,2,34,13,1,0,0,1,38,18,2

%N 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].

%e The irregular triangle begins:

%e 1;

%e 0;

%e 0, 1;

%e 0, 2;

%e 0, 1, 1;

%e 0, 0, 2;

%e 0, 0, 3;

%e 0, 0, 4, 1;

%e 0, 0, 5, 1;

%e 0, 0, 7, 2;

%e 0, 0, 7, 3;

%e 0, 0, 8, 5;

%e 0, 0, 8, 6, 1;

%e ...

%e T(11,3) = 5 since we have: 1+2+8, 1+3+7, 1+4+6, 2+3+6, 2+4+5.

%t Flatten[Table[Length[Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@# &&Length[#]==k&]], {n, 23}, {k, Floor[(Sqrt[8n+1]-1)/2]}]]

%Y Cf. A000290, A003056 (row lengths), A072574, A216652, A362208 (compositions).

%K nonn,tabf

%O 1,8

%A _Stefano Spezia_, Apr 11 2023