A303300 Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by two, including the partition n, and the first element of column k is in row k^2.

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0

Offset

1

Comments

T(n,k) is 0 or 1, so T(n,k) represents the "existence" of the mentioned partition: 1 = exists, 0 = does not exist.

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 1's interleaved with k-1 zeros, and the first element of column k is in row k^2.

From Omar E. Pol, Jan 08 2019: (Start)

Theorem: Let T(n,k) be an irregular triangle read by rows in which column k lists 1's interleaved with k-1 zeros, and the first element of column k is where the row number equals the k-th (m+2)-gonal number, with n >= 1, k >= 1, m >= 0. T(n,k) is also the number of partitions of n into k consecutive parts that differ by m, including the partition n.

About the above theorem, this is the case for m = 2. For m = 1 see the triangle A237048, in which row sums give A001227. For m = 0 see the triangle A051731, in which row sums give A000005. Note that there are infinitely many triangles of this kind, with m >= 0. Also, every triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve. (End)

Links

Table of n, a(n) for n=1..110.

Example

Triangle begins (rows 1..25):
1;
1;
1;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1;
1, 0, 1;
1, 1, 0;
1, 0, 0;
1, 1, 1;
1, 0, 0;
1, 1, 0;
1, 0, 1;
1, 1, 0, 1;
1, 0, 0, 0;
1, 1, 1, 0;
1, 0, 0, 0;
1, 1, 0, 1;
1, 0, 1, 0;
1, 1, 0, 0;
1, 0, 0, 0;
1, 1, 1, 1;
1, 0, 0, 0, 1;
...
For n = 16 there are three partitions of 16 into consecutive parts that differ by two, 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, 1, 0, 1].

Crossrefs

Row sums give A038548.

Cf. A000005, A001227, A000290, A051731, A139600, A237048.

Sequence in context: A190230 A141679 A276254 * A249865 A152904 A249133

Adjacent sequences: A303297 A303298 A303299 * A303301 A303302 A303303

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 21 2018

Status

approved