A363626 Number of integer compositions of n with weighted alternating sum 0.

1, 0, 0, 1, 1, 0, 2, 5, 7, 8, 14, 38, 64, 87, 174, 373, 649, 1069, 2051, 4091, 7453, 13276, 25260, 48990, 91378, 168890, 321661, 618323, 1169126, 2203649, 4211163, 8085240, 15421171, 29390131, 56382040, 108443047, 208077560, 399310778

Offset

0,7

Comments

We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) * i * y_i.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150 (first 51 terms from Max Alekseyev)

Example

The a(3) = 1 through a(10) = 14 compositions:
  (21)  (121)  .  (42)    (331)     (242)      (63)       (541)
                  (3111)  (1132)    (1331)     (153)      (2143)
                          (2221)    (11132)    (4122)     (3232)
                          (21121)   (12221)    (5211)     (4321)
                          (112111)  (23111)    (13122)    (15112)
                                    (121121)   (14211)    (31231)
                                    (1112111)  (411111)   (42121)
                                               (1311111)  (114112)
                                                          (212122)
                                                          (213211)
                                                          (311221)
                                                          (322111)
                                                          (3111121)
                                                          (21211111)

Mathematica

altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]], {k, 1, Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], altwtsum[#]==0&]], {n, 0, 10}]

Crossrefs

The unweighted version is A138364, ranks A344619.

The version for partitions is A363532, ranks A363621.

A000041 counts integer partitions.

A264034 counts partitions by weighted sum, reverse A358194.

A304818 gives weighted sum of prime indices, reverse A318283.

A316524 gives alternating sum of prime indices, reverse A344616.

A363619 gives weighted alternating sum of prime indices, reverse A363620.

A363624 gives weighted alternating sum of Heinz partition, reverse A363625.

Cf. A008284, A053632, A106529, A261079, A320387, A360672, A360675, A362559, A363622, A363623.

Sequence in context: A005624 A275410 A139481 * A340326 A188341 A238364

Adjacent sequences: A363623 A363624 A363625 * A363627 A363628 A363629

Keyword

nonn

Author

Gus Wiseman, Jun 16 2023

Extensions

Terms a(22) onward from Max Alekseyev, Sep 05 2023

Status

approved