A337583 Irregular triangle read by rows: T(n, k) is the number of integer multisets (partitions) that match the multiplicity multiset of exactly k partitions of n.

1, 1, 2, 3, 5, 3, 2, 9, 1, 7, 1, 2, 12, 2, 2, 12, 2, 2, 2, 15, 3, 3, 3, 15, 5, 3, 0, 3, 0, 1, 26, 8, 2, 1, 2, 0, 1, 1, 23, 7, 2, 4, 1, 3, 0, 1, 0, 0, 1, 28, 9, 4, 5, 2, 2, 2, 0, 0, 1, 1, 33, 11, 3, 4, 2, 3, 3, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 45, 10, 8, 4, 4, 1, 4, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 40, 18, 7, 3, 5

Offset

0,3

Links

Álvar Ibeas, Rows until n=66, flattened

Álvar Ibeas, Rows until n=19

Formula

Sum_{k >= 1} k * T(n, k) = A000041(n).

Example

T(5, 1) = 3, T(5, 2) = 2: The partitions of 5 present A088887(5) = 5 different multiplicity multisets. Three of them are attained by a single partition of 5 (for instance, (3, 1) comes from (2, 1, 1, 1) only), whereas (1, 1) and (2, 1) arise from two partitions of 5 each (namely, (4, 1) and (3, 2) for the first and (3, 1, 1) and (2, 2, 1) for the second).
Triangle begins:
  k:  1 2 3 4
      -------
n=0:  1
n=1:  1
n=2:  2
n=3:  3
n=4:  5
n=5:  3 2
n=6:  9 1
n=7:  7 1 2
n=8: 12 2 2
n=9: 12 2 2 2

Crossrefs

Cf. A088887 (row sums), A337587 (row lengths).

Sequence in context: A340641 A322235 A346477 * A172984 A072751 A251542

Adjacent sequences: A337580 A337581 A337582 * A337584 A337585 A337586

Keyword

nonn,tabf

Author

Álvar Ibeas, Sep 02 2020

Status

approved