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