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A035544
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Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).
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13
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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
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OFFSET
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0,5
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COMMENTS
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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)
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LINKS
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Table of n, a(n) for n=0..63.
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Olivier Gérard
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EXTENSIONS
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More terms from David W. Wilson
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STATUS
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approved
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