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

Author

Gus Wiseman, Jun 10 2023

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Total[Accumulate[#]],Total[Accumulate[Reverse[#]]]]&]],{n,30}]