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A301854 Number of positive special sums of integer partitions of n.

%I A301854 #14 Sep 26 2023 14:58:54
%S A301854 1,3,7,13,25,40,67,100,158,220,336,452,649,862,1228,1553,2155,2738,
%T A301854 3674,4612,6124,7497,9857,12118,15524,18821,24152,28863,36549,44002,
%U A301854 54576,65125,80943,95470,117991,139382,169389,199144,242925,283353,342139,400701,479001
%N A301854 Number of positive special sums of integer partitions of n.
%C A301854 A positive special sum of an integer partition y is a number n > 0 such that exactly one submultiset of y sums to n.
%e A301854 The a(4) = 13 special positive subset-sums:
%e A301854 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111),
%e A301854 1<=(211),  3<=(211),  4<=(211),
%e A301854 1<=(31),   3<=(31),   4<=(31),
%e A301854 2<=(22),   4<=(22),
%e A301854 4<=(4).
%t A301854 uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
%t A301854 Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]
%o A301854 (Python)
%o A301854 from collections import Counter
%o A301854 from sympy.utilities.iterables import partitions, multiset_combinations
%o A301854 def A301854(n): return sum(sum(1 for r in Counter(sum(q) for l in range(1,len(p)+1) for q in multiset_combinations(p,l)).values() if r==1) for p in (tuple(Counter(x).elements()) for x in partitions(n))) # _Chai Wah Wu_, Sep 26 2023
%Y A301854 Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854, A301855, A301856.
%K A301854 nonn
%O A301854 1,2
%A A301854 _Gus Wiseman_, Mar 27 2018
%E A301854 a(21)-a(35) from _Alois P. Heinz_, Apr 08 2018
%E A301854 a(36)-a(43) from _Chai Wah Wu_, Sep 26 2023