

A035544


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


13



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

Also the number of integer partitions of n whose skewalternating sum is 0, where we define the skewalternating 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
The reverse version is A357640 (aerated).
(End)


LINKS



EXAMPLE

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.
These partitions are ranked by A357707.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



