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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.
2
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
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.
Sequence in context: A318636 A366975 A267582 * A139127 A239143 A131319
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 28 2023
EXTENSIONS
Terms a(31) and beyond from Andrew Howroyd, Jan 30 2023
STATUS
approved