A363532 Number of integer partitions of n with weighted alternating sum 0.

1, 0, 0, 1, 0, 0, 2, 2, 0, 3, 3, 3, 5, 5, 10, 12, 7, 14, 25, 18, 22, 48, 48, 41, 67, 82, 89, 111, 140, 170, 220, 214, 264, 392, 386, 436, 623, 693, 756, 934, 1102, 1301, 1565, 1697, 2132, 2616, 2727, 3192, 4062, 4550, 5000, 6132, 7197, 8067, 9338, 10750, 12683

Offset: 0

Author

Gus Wiseman, Jun 14 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(11) = 3 through a(15) = 12 partitions (A = 10):
  (33221)   (84)        (751)       (662)       (A5)
  (44111)   (6222)      (5332)      (4442)      (8322)
  (222221)  (7311)      (6421)      (5531)      (9411)
            (621111)    (532111)    (43331)     (722211)
            (51111111)  (42211111)  (54221)     (831111)
                                    (65111)     (3322221)
                                    (432221)    (3333111)
                                    (443111)    (4422111)
                                    (32222111)  (5511111)
                                    (33311111)  (22222221)
                                                (72111111)
                                                (6111111111)
		

Crossrefs

The unweighted version is A035363.

These partitions have ranks A363621.

The triangle for this rank statistic is A363623, reverse A363622.

The version for compositions is A363626.

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.

Programs

Mathematica

altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];
Table[Length[Select[IntegerPartitions[n],altwtsum[#]==0&]],{n,0,30}]