A363528 Number of strict integer partitions of n with weighted sum divisible by reverse-weighted sum.

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.

Links

Table of n, a(n) for n=1..69.

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.

Cf. A008284, A053632, A067538, A222855, A222970, A358137, A359754, A359755, A362558, A362559, A362560.

Sequence in context: A095374 A352063 A300650 * A300649 A086599 A371209

Adjacent sequences: A363525 A363526 A363527 * A363529 A363530 A363531

Keyword

nonn

Author

Gus Wiseman, Jun 10 2023

Status

approved