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

Author

Gus Wiseman, Jan 28 2023

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

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.

Programs

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}]

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

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

Extensions

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