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%I #7 Jun 19 2023 22:35:35
%S 1,0,0,0,0,1,0,0,1,1,1,1,1,1,3,4,4,6,8,7,10,13,13,21,25,24,37,39,40,
%T 58,63,72,94,106,118,144,165,181,224,256,277,341,387,417,504,560,615,
%U 743,818,899,1066,1171,1285,1502,1655,1819,2108,2315,2547,2915
%N Number of integer partitions of n with weighted sum 3*n.
%C Are the partitions counted all of length > 4?
%C The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.
%e The partition (2,2,1,1,1,1) has sum 8 and weighted sum 24 so is counted under a(8).
%e The a(13) = 1 through a(18) = 8 partitions:
%e (332221) (333221) (33333) (442222) (443222) (443331)
%e (4322111) (522222) (5322211) (4433111) (444222)
%e (71111111) (4332111) (55111111) (5332211) (533322)
%e (63111111) (63211111) (55211111) (4443111)
%e (63311111) (7222221)
%e (72221111) (55311111)
%e (64221111)
%e (A11111111)
%t Table[Length[Select[IntegerPartitions[n],Total[Accumulate[Reverse[#]]]==3n&]],{n,0,30}]
%Y The version for compositions is A231429.
%Y The reverse version is A363526.
%Y These partitions have ranks A363531.
%Y A000041 counts integer partitions, strict A000009.
%Y A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
%Y A264034 counts partitions by weighted sum, reverse A358194.
%Y A304818 gives weighted sum of prime indices, row-sums of A359361.
%Y A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
%Y A320387 counts multisets by weighted sum, zero-based A359678.
%Y Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528, A363532.
%K nonn
%O 0,15
%A _Gus Wiseman_, Jun 11 2023