A360071 Regular tetrangle where T(n,k,i) = number of integer partitions of n of length k with i distinct parts.

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset

1,23

Comments

I call this a tetrangle because it is a sequence of finite triangles. - Gus Wiseman, Jan 30 2023

Links

Table of n, a(n) for n=1..84.

Example

Tetrangle begins:
  1   1     1       1         1           1             1
      1 0   0 1     1 1       0 2         1 2           0 3
            1 0 0   0 1 0     0 2 0       1 1 1         0 3 1
                    1 0 0 0   0 1 0 0     0 2 0 0       0 2 1 0
                              1 0 0 0 0   0 1 0 0 0     0 2 0 0 0
                                          1 0 0 0 0 0   0 1 0 0 0 0
                                                        1 0 0 0 0 0 0
For example, finite triangle n = 5 counts the following partitions:
    (5)
     .    (41)(32)
     .   (311)(221)  .
     .     (2111)    .   .
  (11111)     .      .   .   .

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]], {n, 1, 9}, {k, 1, n}, {i, 1, k}]

Crossrefs

Row sums are A008284 (partitions by number of parts), reverse A058398.

First columns i = 1 are A051731.

Last columns i = k are A060016.

Column sums are A116608 (partitions by number of distinct parts).

Positive terms are counted by A360072.

A000041 counts partitions, strict A000009.

Other tetrangles: A318393, A318816, A320808, A334433, A345197.

Cf. A055884, A327482, A331195, A360010, A360069.

Sequence in context: A300824 A269248 A092078 * A358724 A325135 A263577

Adjacent sequences: A360068 A360069 A360070 * A360072 A360073 A360074

Keyword

nonn,tabf

Author

Gus Wiseman, Jan 28 2023

Status

approved