This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325835 #9 May 31 2019 05:33:58
%S A325835 0,0,1,2,3,5,9,10,14,22,30,33,46,52,74,107,101,123,171,182,225
%N A325835 Number of integer partitions of 2*n having one more distinct submultiset than distinct subset-sums.
%C A325835 The number of submultisets of a partition is the product of its multiplicities, each plus one. A subset-sum of an integer partition is the sum of some submultiset of its parts. These are partitions with one subset-sum which is the sum of two distinct submultisets, while all others are the sum of only one submultiset.
%C A325835 The Heinz numbers of these partitions are given by A325802.
%e A325835 The a(2) = 1 through a(8) = 14 partitions:
%e A325835 (211) (321) (422) (532) (633) (743) (844)
%e A325835 (3111) (431) (541) (642) (752) (853)
%e A325835 (41111) (5221) (651) (761) (862)
%e A325835 (5311) (4332) (7322) (871)
%e A325835 (511111) (5331) (7331) (5443)
%e A325835 (6222) (7421) (7441)
%e A325835 (6411) (7511) (7531)
%e A325835 (33222) (72221) (8332)
%e A325835 (6111111) (74111) (8521)
%e A325835 (71111111) (8611)
%e A325835 (82222)
%e A325835 (83311)
%e A325835 (85111)
%e A325835 (811111111)
%e A325835 For example, the partition (7,5,3,1) has submultisets (), (1), (3), (5), (7), (3,1), (5,1), (5,3), (7,1), (7,3), (7,5), (5,3,1), (7,3,1), (7,5,1), (7,5,3), (7,5,3,1), all of which have different sums except for (5,3) and (7,1), which both sum to 8, so (7,5,3,1) is counted under a(8).
%t A325835 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==1+Length[Union[Total/@Subsets[#]]]&]],{n,0,20,2}]
%Y A325835 Cf. A002033, A088880, A088881, A108917, A126796, A276024, A325799, A325800, A325801, A325802, A325828, A325830, A325833, A325836.
%K A325835 nonn,more
%O A325835 0,4
%A A325835 _Gus Wiseman_, May 29 2019