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A366527
Number of integer partitions of 2n containing at least one even part.
3
0, 1, 3, 7, 16, 32, 62, 113, 199, 339, 563, 913, 1453, 2271, 3496, 5308, 7959, 11798, 17309, 25151, 36225, 51748, 73359, 103254, 144363, 200568, 277007, 380437, 519715, 706412, 955587, 1286762, 1725186, 2303388, 3063159, 4058041, 5356431, 7045454, 9235841
OFFSET
0,3
COMMENTS
Also partitions of 2n with even product.
FORMULA
a(n) = A000041(2n) - A000009(2n).
EXAMPLE
The a(1) = 1 through a(4) = 16 partitions:
(2) (4) (6) (8)
(22) (42) (44)
(211) (222) (62)
(321) (332)
(411) (422)
(2211) (431)
(21111) (521)
(611)
(2222)
(3221)
(4211)
(22211)
(32111)
(41111)
(221111)
(2111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[2n], Or@@EvenQ/@#&]], {n, 0, 15}]
CROSSREFS
This is the even bisection of A047967.
For odd instead of even parts we have A182616, ranks A366321 or A366528.
These partitions have ranks A366529, subset of A324929.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions w/ at least one odd and even part, ranks A366532.
A086543 counts partitions of n not containing n/2, ranks A366319.
A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
Sequence in context: A179904 A376334 A298311 * A161810 A318604 A084631
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 16 2023
STATUS
approved