A337586 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) for which the number of partitions of n with matching multiplicity multiset is odd (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 3, 2, 1, 1, 1, 0, 3, 0, 3, 3, 2, 1, 1, 1, 1, 0, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 3, 2, 1, 5, 3, 2, 1, 1, 1, 2, 2, 1, 3, 7, 3, 5, 3, 2, 1, 1, 1, 0, 0, 2, 2, 2, 5, 3, 5, 3, 2, 1, 1, 1, 1, 0, 3, 3, 4, 5, 5, 3, 5, 3

Offset: 1

Author

Álvar Ibeas, Sep 02 2020

Keywords

• nonn
• tabl

Comments

The relevant partitions of n have exactly k parts.

The number of multiplicity multisets of size k met by a positive even number of partitions of n is A337584(n, k) - T(n, k).

Examples

			The 3 = A008284(6, 2) partitions of 6 into 2 parts show 2 = A337584(6, 2) different multiplicity multisets: (1, 1) is attained by two of those partitions ((5, 1) and (4, 2)) and the other (2) just by one, (3, 3). Then, T(6, 2) = 1.
Triangle begins:
  k:  1 2 3 4 5 6 7 8 9 10
      --------------------
n=1:  1
n=2:  1 1
n=3:  1 1 1
n=4:  1 2 1 1
n=5:  1 0 0 1 1
n=6:  1 1 3 2 1 1
n=7:  1 1 2 1 2 1 1
n=8:  1 2 1 3 3 2 1 1
n=9:  1 0 3 0 3 3 2 1 1
n=10: 1 1 0 3 3 3 3 2 1 1
		

Links

• Álvar Ibeas, First 72 rows, flattened
• Álvar Ibeas, First 30 rows

Crossrefs

Cf. A008284, A337585 (row sums), A337584.

Formula

T(n, k) == A008284(n, k) (mod 2).

If k > (2*n+1)/3, T(n, k) = A337585(n - k).