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A337584 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) that match the multiplicity multiset of some partition of n (n >= 1, 1 <= k <= n). 2
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 4, 3, 2, 1, 1, 1, 1, 3, 2, 4, 3, 2, 1, 1, 1, 2, 2, 4, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 3, 5, 3, 2, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 2, 2, 4, 5, 7, 8, 8, 5, 5, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

The relevant partitions of n have exactly k parts.

Column k is k-periodic from n = k*(k+1)/2.

LINKS

Álvar Ibeas, First 71 rows, flattened

Álvar Ibeas, First 30 rows

FORMULA

If k > (2*n+1)/3, T(n, k) = A088887(n - k).

If n >= k*(k+1)/2, T(n, k) = Sum_{d | gcd(n, k)} A000837(k/d).

T(n, k) = A000041(k) iff k|n and n >= k*(k+1)/2.

EXAMPLE

There is no partition of 5 with multiplicity multiset (3) or (1, 1, 1).

Indeed, both (2 = A008284(5, 3)) partitions of 5 into 3 parts (namely, (3, 1, 1) and (2, 2, 1)) have multiplicities (2, 1). Therefore, T(5, 3) = 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 1 1 1 1

n=6:  1 2 3 2 1 1

n=7:  1 1 2 2 2 1 1

n=8:  1 2 2 4 3 2 1 1

n=9:  1 1 3 2 4 3 2 1 1

n=10: 1 2 2 4 4 4 3 2 1 1

CROSSREFS

Cf. A000041, A008284, A088887 (row sums).

Sequence in context: A261794 A328929 A098744 * A273975 A025429 A325561

Adjacent sequences:  A337581 A337582 A337583 * A337585 A337586 A337587

KEYWORD

nonn,tabl

AUTHOR

Álvar Ibeas, Sep 02 2020

STATUS

approved

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Last modified January 18 00:56 EST 2021. Contains 340249 sequences. (Running on oeis4.)