A035544 Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).

1, 0, 1, 0, 3, 0, 4, 0, 10, 0, 13, 0, 28, 0, 37, 0, 72, 0, 96, 0, 172, 0, 230, 0, 391, 0, 522, 0, 846, 0, 1129, 0, 1766, 0, 2348, 0, 3564, 0, 4722, 0, 6992, 0, 9226, 0, 13371, 0, 17568, 0, 25006, 0, 32708, 0, 45817, 0, 59668, 0, 82430, 0, 106874, 0, 145830, 0, 188260, 0

Offset

0,5

Comments

From Gus Wiseman, Oct 12 2022: (Start)

Also the number of integer partitions of n whose skew-alternating sum is 0, where we define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ... These are the conjugates of the partitions described in the name. For example, the a(0) = 1 through a(8) = 10 partitions are:

() . (11) . (22) . (33) . (44)

(211) (321) (422)

(1111) (2211) (431)

(111111) (2222)

(3221)

(3311)

(22211)

(221111)

(2111111)

(11111111)

The ordered version (compositions) is A138364

These partitions are ranked by A357636, reverse A357632.

The reverse version is A357640 (aerated).

Cf. A357623, A357629, A357630, A357634, A357646, A357705.

(End)

Links

Table of n, a(n) for n=0..63.

Example

From Gus Wiseman, Oct 12 2022: (Start)
The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (2)  .  (4)   .  (6)    .  (8)
                 (22)     (42)      (44)
                 (31)     (222)     (53)
                          (321)     (62)
                                    (71)
                                    (422)
                                    (431)
                                    (2222)
                                    (3221)
                                    (3311)
(End)

Mathematica

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[n], skats[#]==0&]], {n, 0, 30}] (* Gus Wiseman, Oct 12 2022 *)

Crossrefs

The case with at least one odd part is A035550.

Removing zeros gives A035594.

Central column k=0 of A357638.

These partitions are ranked by A357707.

A000041 counts integer partitions.

A344651 counts partitions by alternating sum, ordered A097805.

Cf. A035363, A053251, A298311, A357189, A357487, A357488.

Sequence in context: A237558 A060034 A308216 * A129718 A127375 A238573

Adjacent sequences: A035541 A035542 A035543 * A035545 A035546 A035547

Keyword

nonn

Author

Olivier Gérard

Extensions

More terms from David W. Wilson

Status

approved