cp's OEIS Frontend

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 A362221 #9 Apr 16 2023 20:37:57
%S A362221 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 A362221 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 A362221 4,23,7,0,0,3,27,10,1,0,0,2,34,13,1,0,0,1,38,18,2
%N 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].
%e A362221 The irregular triangle begins:
%e A362221     1;
%e A362221     0;
%e A362221     0, 1;
%e A362221     0, 1;
%e A362221     0, 2;
%e A362221     0, 1, 1;
%e A362221     0, 1, 1;
%e A362221     0, 0, 2;
%e A362221     0, 0, 3;
%e A362221     0, 0, 4, 1;
%e A362221     0, 0, 5, 1;
%e A362221     0, 0, 7, 2;
%e A362221     0, 0, 7, 3;
%e A362221     0, 0, 8, 5;
%e A362221     0, 0, 8, 6, 1;
%e A362221     ...
%e A362221 T(11,3) = 5 since we have: 1+2+8, 1+3+7, 1+4+6, 2+3+6, 2+4+5.
%t A362221 Flatten[Table[Length[Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@# &&Length[#]==k&]], {n, 23}, {k, Floor[(Sqrt[8n+1]-1)/2]}]]
%Y A362221 Cf. A000290, A003056 (row lengths), A072574, A216652, A362208 (compositions).
%K A362221 nonn,tabf
%O A362221 1,8
%A A362221 _Stefano Spezia_, Apr 11 2023