A304796 Number of special sums of integer partitions of n.

1, 2, 5, 10, 18, 32, 51, 82, 122, 188, 262, 392, 529, 750, 997, 1404, 1784, 2452, 3123, 4164, 5239, 6916, 8499, 11112, 13693, 17482, 21257, 27162, 32581, 41114, 49606, 61418, 73474, 91086, 107780, 132874, 157359, 191026, 225159, 274110, 320691, 386722, 453875

Offset

0,2

Comments

A special sum of an integer partition y is a number n >= 0 such that exactly one submultiset of y sums to n.

Links

Table of n, a(n) for n=0..42.

Formula

a(n) = A301854(n) + A000041(n).

Example

The a(4) = 18 special positive subset-sums:
0<=(4), 4<=(4),
0<=(22), 2<=(22), 4<=(22),
0<=(31), 1<=(31), 3<=(31), 4<=(31),
0<=(211), 1<=(211), 3<=(211), 4<=(211),
0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).

Mathematica

uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]], Total], Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]], {n, 25}]

Crossrefs

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854.

Sequence in context: A034350 A006327 A185721 * A103577 A326508 A079006

Adjacent sequences: A304793 A304794 A304795 * A304797 A304798 A304799

Keyword

nonn

Author

Gus Wiseman, May 18 2018

Extensions

More terms from Alois P. Heinz, May 18 2018

a(36)-a(42) from Chai Wah Wu, Sep 26 2023

Status

approved