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%I #17 Sep 22 2023 05:23:12
%S 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,
%T 122,151,161,195,201,247,266,312,330,386,419,487,520,600,641,742,793,
%U 901,979,1088,1186,1331,1454,1605,1730,1925,2102,2311,2525,2741,3001
%N a(n) = number of partitions of n whose sum multiset is free of duplicates; see Comments.
%C 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}.
%F a(n) = A325877(n) - (1 - n mod 2) for n > 0. - _Andrew Howroyd_, Sep 17 2023
%e 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].
%t s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}];
%t g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Total, s[n, k]]];
%t t[n_] := Table[g[n, k], {k, 1, PartitionsP[n]}];
%t a[n_] := Count[t[n], True]
%t Table[a[n], {n, 1, 40}]
%Y Cf. A000041, A236912, A325877, A363994.
%K nonn
%O 0,3
%A _Clark Kimberling_, Sep 17 2023
%E More terms from _Alois P. Heinz_, Sep 17 2023
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