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%I #8 Nov 08 2023 07:50:40
%S 0,0,0,0,1,1,4,4,10,13,23,27,52,60,94,118,175,213,310,373,528,643,862,
%T 1044,1403,1699,2199,2676,3426,4131,5256,6295,7884,9479,11722,14047,
%U 17296,20623,25142,29942,36299,43081,51950,61439,73668,87040,103748,122149,145155,170487
%N Number of non-knapsack integer partitions of n.
%C A multiset is non-knapsack if there exist two different submultisets with the same sum.
%F a(n) = A000041(n) - A108917(n).
%e The a(4) = 1 through a(9) = 13 partitions:
%e (211) (2111) (321) (3211) (422) (3321)
%e (2211) (22111) (431) (4221)
%e (3111) (31111) (3221) (4311)
%e (21111) (211111) (4211) (5211)
%e (22211) (32211)
%e (32111) (33111)
%e (41111) (42111)
%e (221111) (222111)
%e (311111) (321111)
%e (2111111) (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%t Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Total/@Union[Subsets[#]]&]], {n,0,15}]
%Y The complement is counted by A108917, strict A275972, ranks A299702.
%Y These partitions have ranks A299729.
%Y The strict case is A316402.
%Y The binary version is A366753, ranks A366740.
%Y A000041 counts integer partitions, strict A000009.
%Y A276024 counts positive subset-sums of partitions, strict A284640.
%Y A304792 counts subset-sum of partitions, strict A365925.
%Y A365543 counts partitions with subset-sum k, complement A046663.
%Y A365661 counts strict partitions with subset-sum k, complement A365663.
%Y A366738 counts semi-sums of partitions, strict A366741.
%Y Cf. A002033, A006827, A122768, A126796, A238628, A365923, A365924, A367095.
%K nonn
%O 0,7
%A _Gus Wiseman_, Nov 08 2023