A364613 a(n) = number of partitions of n whose sum multiset is free of duplicates; see Comments.

1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 18, 20, 26, 29, 36, 38, 50, 53, 67, 69, 89, 95, 115, 122, 151, 161, 195, 201, 247, 266, 312, 330, 386, 419, 487, 520, 600, 641, 742, 793, 901, 979, 1088, 1186, 1331, 1454, 1605, 1730, 1925, 2102, 2311, 2525, 2741, 3001

Offset

0,3

Comments

If M is a multiset of real numbers, then the sum multiset of M consists of the sums of pairs of distinct numbers in M. For example, the sum multiset of (1,2,4,5) is {3,5,6,6,7,9}.

Links

Table of n, a(n) for n=0..56.

Formula

a(n) = A325877(n) - (1 - n mod 2) for n > 0. - Andrew Howroyd, Sep 17 2023

Example

The partitions of 8 are [8], [7,1], [6,2], [6,1,1], [5,3], [5,2,1], [5,1,1,1], [4,4], [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,3,2], [3,3,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1]. The 7 partitions whose sum multiset is duplicate-free are [8], [7,1], [6,2], [5,3], [5,2,1], [4,4], [4,3,1].

Mathematica

s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}];
g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Total, s[n, k]]];
t[n_] := Table[g[n, k], {k, 1, PartitionsP[n]}];
a[n_] := Count[t[n], True]
Table[a[n], {n, 1, 40}]

Crossrefs

Cf. A000041, A236912, A325877, A363994.

Sequence in context: A185226 A096765 A025147 * A032230 A238789 A126793

Adjacent sequences: A364610 A364611 A364612 * A364614 A364615 A364616

Keyword

nonn

Author

Clark Kimberling, Sep 17 2023

Extensions

More terms from Alois P. Heinz, Sep 17 2023

Status

approved