cp's OEIS Frontend

A366738 Number of semi-sums of integer partitions of n.

%I A366738 #8 Nov 06 2023 22:58:42
%S A366738 0,0,1,2,5,9,17,28,46,72,111,166,243,352,500,704,973,1341,1819,2459,
%T A366738 3277,4363,5735,7529,9779,12685,16301,20929,26638,33878,42778,53942,
%U A366738 67583,84600,105270,130853,161835,199896,245788,301890,369208,451046,549002,667370
%N A366738 Number of semi-sums of integer partitions of n.
%C A366738 We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
%e A366738 The partitions of 6 and their a(6) = 17 semi-sums:
%e A366738        (6) ->
%e A366738       (51) -> 6
%e A366738       (42) -> 6
%e A366738      (411) -> 2,5
%e A366738       (33) -> 6
%e A366738      (321) -> 3,4,5
%e A366738     (3111) -> 2,4
%e A366738      (222) -> 4
%e A366738     (2211) -> 2,3,4
%e A366738    (21111) -> 2,3
%e A366738   (111111) -> 2
%t A366738 Table[Total[Length[Union[Total/@Subsets[#,{2}]]]&/@IntegerPartitions[n]],{n,0,15}]
%Y A366738 The non-binary version is A304792.
%Y A366738 The strict non-binary version is A365925.
%Y A366738 For prime indices instead of partitions we have A366739.
%Y A366738 The strict case is A366741.
%Y A366738 A000041 counts integer partitions, strict A000009.
%Y A366738 A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A366738 A126796 counts complete partitions, ranks A325781, strict A188431.
%Y A366738 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A366738 A365924 counts incomplete partitions, ranks A365830, strict A365831.
%Y A366738 Cf. A008967, A046663, A117855, A122768, A238628, A299701, A365543, A365544, A366753, A367095.
%K A366738 nonn
%O A366738 0,4
%A A366738 _Gus Wiseman_, Nov 06 2023
%E A366738 More terms from _Alois P. Heinz_, Nov 06 2023