A330466 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 2, n >= 1, k >= 1, and the first element of column k is in row k^2.

1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 0, 1, 2, 0, 4, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 2, 3, 4, 1, 0, 0, 0, 5, 1, 2, 0, 0, 0, 1, 0, 3, 0, 0, 1, 2, 0, 4, 0, 1, 0, 0, 0, 0, 1, 2, 3, 0, 5, 1, 0, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Apr 30 2020

Keywords

• nonn
• tabf

Comments

Since the trivial partition n is counted, so T(n,1) = 1.

This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in row k^2.

Conjecture: row sums give A066839.

Examples

			Triangle begins (rows 1..25):
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0;
1, 2, 0;
1, 0, 3;
1, 2, 0, 4;
1, 0, 0, 0;
1, 2, 3, 0;
1, 0, 0, 0;
1, 2, 0, 4;
1, 0, 3, 0;
1, 2, 0, 0;
1, 0, 0, 0;
1, 2, 3, 4;
1, 0, 0, 0, 5;
...
For n = 16 there are three partitions of 16 into consecutive parts that differ by 2, including 16 as a partition. They are [16], [9, 7] and [7, 5, 3, 1]. The number of parts of these partitions are 1, 2 and 4 respectively, so the 16th row of the triangle is [1, 2, 0, 4].
		

Crossrefs

Cf. A000290, A066839, A237048, A303300.

Other triangles of the same family are A127093 and A285914.

Formula

T(n,k) = k*A303300(n,k).