cp's OEIS Frontend

A363526 Number of integer partitions of n with reverse-weighted sum 3*n.

%I A363526 #10 Jun 12 2023 08:44:13
%S A363526 1,0,0,0,0,1,1,1,1,1,2,1,2,2,2,4,3,2,4,4,4,5,5,4,7,7,5,8,7,6,11,9,8,
%T A363526 11,10,10,13,12,11,15,15,12,17,16,14,20,18,16,22,20,19,24,22,20,27,26,
%U A363526 23,29,27,25,33,30,28,35,33,31,38,36,33,41,40
%N A363526 Number of integer partitions of n with reverse-weighted sum 3*n.
%C A363526 Are the partitions counted all of length 4 or 5?
%C A363526 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 A363526 The partition (6,4,4,1) has sum 15 and reverse-weighted sum 45 so is counted under a(15).
%e A363526 The a(n) partitions for n = {5, 10, 15, 16, 21, 24}:
%e A363526   (1,1,1,1,1)  (4,3,2,1)    (6,4,4,1)    (6,5,4,1)  (8,6,6,1)   (9,7,7,1)
%e A363526                (2,2,2,2,2)  (6,5,2,2)    (6,6,2,2)  (8,7,4,2)   (9,8,5,2)
%e A363526                             (7,3,3,2)    (7,4,3,2)  (9,5,5,2)   (9,9,3,3)
%e A363526                             (3,3,3,3,3)             (9,6,3,3)   (10,6,6,2)
%e A363526                                                     (10,4,4,3)  (10,7,4,3)
%e A363526                                                                 (11,5,5,3)
%e A363526                                                                 (12,4,4,4)
%t A363526 Table[Length[Select[IntegerPartitions[n],Total[Accumulate[#]]==3n&]],{n,0,30}]
%Y A363526 Positions of terms with omega > 4 appear to be A079998.
%Y A363526 The version for compositions is A231429.
%Y A363526 The non-reverse version is A363527.
%Y A363526 These partitions have ranks A363530, reverse A363531.
%Y A363526 A000041 counts integer partitions, strict A000009.
%Y A363526 A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
%Y A363526 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363526 A304818 gives weighted sum of prime indices, row-sums of A359361.
%Y A363526 A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
%Y A363526 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A363526 Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528.
%K A363526 nonn
%O A363526 0,11
%A A363526 _Gus Wiseman_, Jun 10 2023