%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