A360072 Number of pairs of positive integers (k,i) such that k >= i and there exists an integer partition of n of length k with i distinct parts.

0, 1, 2, 3, 5, 5, 9, 9, 13, 14, 18, 19, 26, 25, 30, 34, 39, 40, 48, 48, 56, 59, 64, 67, 78, 78, 84, 89, 97, 99, 111, 111, 121, 125, 131, 137, 149, 149, 158, 165, 176, 177, 190, 191, 202, 210, 216, 222, 238, 239, 250, 256, 266, 270, 284, 289, 302, 307, 316, 323

Offset

0,3

Comments

This is the number of nonzero terms in the n-th triangle of A360071.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = A000005(n) + Sum_{k=2..floor((sqrt(8*n+1)-1)/2)} (1 + n - binomial(k+1,2)) for n > 0. - Andrew Howroyd, Jan 30 2023

Example

The a(5) = 5 pairs are: (1,1), (2,2), (3,2), (4,2), (5,1). The pair (3,3) is absent because it is not possible to partition 5 into 3 parts, all 3 of which are distinct.
The a(6) = 9 pairs are: (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,2), (5,2), (6,1). The pair (3,3) is present because (3,2,1) is a partition of 6 into 3 parts, all 3 of which are distinct.

Mathematica

Table[Count[Flatten[Sign[Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]], {k, 1, n}, {i, 1, k}]]], 1], {n, 0, 30}]

Prog

(PARI) a(n) = if(n < 1, 0, numdiv(n) + sum(k=2, (sqrtint(8*n+1)-1)\2, n-binomial(k+1, 2)+1)) \\ Andrew Howroyd, Jan 30 2023

Crossrefs

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts, reverse A058398.

A116608 counts partitions by number of distinct parts.

Cf. A000005, A051731, A055884, A060016, A331195, A360010, A360071.

Sequence in context: A318636 A366975 A267582 * A139127 A239143 A131319

Adjacent sequences: A360069 A360070 A360071 * A360073 A360074 A360075

Keyword

nonn

Author

Gus Wiseman, Jan 28 2023

Extensions

Terms a(31) and beyond from Andrew Howroyd, Jan 30 2023

Status

approved