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A363528
Number of strict integer partitions of n with weighted sum divisible by reverse-weighted sum.
3
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 2, 3, 9, 3, 4, 11, 4, 5, 16, 6, 8, 24, 8, 10, 31, 11, 14, 41, 18, 18, 59, 21, 27, 74, 30, 32, 100, 35, 43, 128, 54, 53, 173, 58, 78, 215, 81, 88, 294, 97, 123, 362, 150, 146, 469, 162, 221, 577
OFFSET
1,12
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.
EXAMPLE
The a(n) partitions for n = 1, 12, 15, 21, 24, 26:
(1) (12) (15) (21) (24) (26)
(9,2,1) (11,3,1) (15,5,1) (17,6,1) (11,8,4,2,1)
(9,3,2,1) (16,3,2) (18,4,2) (12,6,5,2,1)
(11,7,2,1) (12,9,2,1) (13,5,4,3,1)
(12,5,3,1) (13,7,3,1)
(10,5,3,2,1) (14,5,4,1)
(15,4,3,2)
(10,8,3,2,1)
(11,6,4,2,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]], {n, 30}]
CROSSREFS
The non-strict version is A363525.
A000041 counts integer partitions, strict A000009.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
A363526 counts partitions with weighted sum 3n, reverse A363531.
Sequence in context: A095374 A352063 A300650 * A300649 A086599 A371209
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 10 2023
STATUS
approved