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Number of integer partitions of 2n containing at least one even part.
3

%I #7 Oct 16 2023 13:42:33

%S 0,1,3,7,16,32,62,113,199,339,563,913,1453,2271,3496,5308,7959,11798,

%T 17309,25151,36225,51748,73359,103254,144363,200568,277007,380437,

%U 519715,706412,955587,1286762,1725186,2303388,3063159,4058041,5356431,7045454,9235841

%N Number of integer partitions of 2n containing at least one even part.

%C Also partitions of 2n with even product.

%F a(n) = A000041(2n) - A000009(2n).

%e The a(1) = 1 through a(4) = 16 partitions:

%e (2) (4) (6) (8)

%e (22) (42) (44)

%e (211) (222) (62)

%e (321) (332)

%e (411) (422)

%e (2211) (431)

%e (21111) (521)

%e (611)

%e (2222)

%e (3221)

%e (4211)

%e (22211)

%e (32111)

%e (41111)

%e (221111)

%e (2111111)

%t Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]

%Y This is the even bisection of A047967.

%Y For odd instead of even parts we have A182616, ranks A366321 or A366528.

%Y These partitions have ranks A366529, subset of A324929.

%Y A000041 counts integer partitions, strict A000009.

%Y A006477 counts partitions w/ at least one odd and even part, ranks A366532.

%Y A086543 counts partitions of n not containing n/2, ranks A366319.

%Y A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.

%Y Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 16 2023