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Number of special sums of integer partitions of n.
3

%I #13 Sep 26 2023 14:58:01

%S 1,2,5,10,18,32,51,82,122,188,262,392,529,750,997,1404,1784,2452,3123,

%T 4164,5239,6916,8499,11112,13693,17482,21257,27162,32581,41114,49606,

%U 61418,73474,91086,107780,132874,157359,191026,225159,274110,320691,386722,453875

%N Number of special sums of integer partitions of n.

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

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

%e The a(4) = 18 special positive subset-sums:

%e 0<=(4), 4<=(4),

%e 0<=(22), 2<=(22), 4<=(22),

%e 0<=(31), 1<=(31), 3<=(31), 4<=(31),

%e 0<=(211), 1<=(211), 3<=(211), 4<=(211),

%e 0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).

%t uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];

%t Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

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

%K nonn

%O 0,2

%A _Gus Wiseman_, May 18 2018

%E More terms from _Alois P. Heinz_, May 18 2018

%E a(36)-a(42) from _Chai Wah Wu_, Sep 26 2023