cp's OEIS Frontend

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

%I A035544 #19 Oct 13 2022 11:19:29
%S A035544 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,
%T A035544 0,846,0,1129,0,1766,0,2348,0,3564,0,4722,0,6992,0,9226,0,13371,0,
%U A035544 17568,0,25006,0,32708,0,45817,0,59668,0,82430,0,106874,0,145830,0,188260,0
%N A035544 Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).
%C A035544 From _Gus Wiseman_, Oct 12 2022: (Start)
%C A035544 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:
%C A035544   ()  .  (11)  .  (22)    .  (33)      .  (44)
%C A035544                   (211)      (321)        (422)
%C A035544                   (1111)     (2211)       (431)
%C A035544                              (111111)     (2222)
%C A035544                                           (3221)
%C A035544                                           (3311)
%C A035544                                           (22211)
%C A035544                                           (221111)
%C A035544                                           (2111111)
%C A035544                                           (11111111)
%C A035544 The ordered version (compositions) is A138364
%C A035544 These partitions are ranked by A357636, reverse A357632.
%C A035544 The reverse version is A357640 (aerated).
%C A035544 Cf. A357623, A357629, A357630, A357634, A357646, A357705.
%C A035544 (End)
%e A035544 From _Gus Wiseman_, Oct 12 2022: (Start)
%e A035544 The a(0) = 1 through a(8) = 10 partitions:
%e A035544   ()  .  (2)  .  (4)   .  (6)    .  (8)
%e A035544                  (22)     (42)      (44)
%e A035544                  (31)     (222)     (53)
%e A035544                           (321)     (62)
%e A035544                                     (71)
%e A035544                                     (422)
%e A035544                                     (431)
%e A035544                                     (2222)
%e A035544                                     (3221)
%e A035544                                     (3311)
%e A035544 (End)
%t A035544 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A035544 Table[Length[Select[IntegerPartitions[n],skats[#]==0&]],{n,0,30}] (* _Gus Wiseman_,Oct 12 2022 *)
%Y A035544 The case with at least one odd part is A035550.
%Y A035544 Removing zeros gives A035594.
%Y A035544 Central column k=0 of A357638.
%Y A035544 These partitions are ranked by A357707.
%Y A035544 A000041 counts integer partitions.
%Y A035544 A344651 counts partitions by alternating sum, ordered A097805.
%Y A035544 Cf. A035363, A053251, A298311, A357189, A357487, A357488.
%K A035544 nonn
%O A035544 0,5
%A A035544 _Olivier Gérard_
%E A035544 More terms from _David W. Wilson_