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A362559
Number of integer partitions of n whose weighted sum is divisible by n.
11
1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, 14, 18, 25, 28, 26, 42, 47, 52, 73, 77, 100, 118, 122, 158, 188, 219, 266, 313, 367, 412, 489, 578, 698, 809, 914, 1094, 1268, 1472, 1677, 1948, 2305, 2656, 3072, 3527, 4081, 4665, 5342, 6225, 7119, 8150, 9408
OFFSET
1,3
COMMENTS
The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.
EXAMPLE
The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
The a(1) = 1 through a(9) = 5 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(111) (11111) (222) (3211) (3311) (333)
(3111) (1111111) (221111) (4221)
(222111)
(111111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]], n]&]], {n, 30}]
CROSSREFS
For median instead of mean we have A362558.
The complement is counted by A362560.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Sequence in context: A029165 A035431 A008681 * A097242 A183202 A306878
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2023
STATUS
approved