A357707 -id:A357707 - cp's OEIS frontend

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

Olivier Gérard

nonn

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)

			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)

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.

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 *)

More terms from David W. Wilson

cp's OEIS Frontend

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

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