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A357709
Number of integer partitions of n whose length is twice their alternating sum.
2
1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 6, 6, 9, 11, 13, 18, 21, 28, 32, 44, 49, 65, 76, 96, 114, 141, 170, 204, 250, 295, 361, 425, 516, 606, 734, 858, 1031, 1210, 1440, 1690, 2000, 2347, 2759, 3240, 3786, 4441, 5174, 6053, 7030, 8210, 9509, 11074, 12807, 14870
OFFSET
0,9
COMMENTS
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The alternating sum of a partition is also the number of odd conjugate parts.
EXAMPLE
The a(1) = 0 through a(12) = 6 partitions:
. . 21 . 32 3111 43 3221 54 3331 65 4332
4211 411111 4222 422111 4431
4321 521111 5322
5311 5421
6411
51111111
MATHEMATICA
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], Length[#]==2ats[#]&]], {n, 0, 30}]
CROSSREFS
This is the "twice" version of A357189, ranked by A357486.
The version for compositions is A357847.
These partitions are ranked by A357848.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
Sequence in context: A082146 A037145 A341466 * A238790 A238787 A293170
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 16 2022
STATUS
approved