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

Author

Gus Wiseman, Jun 16 2023

Keywords

• nonn

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.

Examples

			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)
		

Links

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

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.

Programs

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}]

Extensions

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