cp's OEIS Frontend

A366527 Number of integer partitions of 2n containing at least one even part.

%I A366527 #7 Oct 16 2023 13:42:33
%S A366527 0,1,3,7,16,32,62,113,199,339,563,913,1453,2271,3496,5308,7959,11798,
%T A366527 17309,25151,36225,51748,73359,103254,144363,200568,277007,380437,
%U A366527 519715,706412,955587,1286762,1725186,2303388,3063159,4058041,5356431,7045454,9235841
%N A366527 Number of integer partitions of 2n containing at least one even part.
%C A366527 Also partitions of 2n with even product.
%F A366527 a(n) = A000041(2n) - A000009(2n).
%e A366527 The a(1) = 1 through a(4) = 16 partitions:
%e A366527   (2)  (4)    (6)      (8)
%e A366527        (22)   (42)     (44)
%e A366527        (211)  (222)    (62)
%e A366527               (321)    (332)
%e A366527               (411)    (422)
%e A366527               (2211)   (431)
%e A366527               (21111)  (521)
%e A366527                        (611)
%e A366527                        (2222)
%e A366527                        (3221)
%e A366527                        (4211)
%e A366527                        (22211)
%e A366527                        (32111)
%e A366527                        (41111)
%e A366527                        (221111)
%e A366527                        (2111111)
%t A366527 Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]
%Y A366527 This is the even bisection of A047967.
%Y A366527 For odd instead of even parts we have A182616, ranks A366321 or A366528.
%Y A366527 These partitions have ranks A366529, subset of A324929.
%Y A366527 A000041 counts integer partitions, strict A000009.
%Y A366527 A006477 counts partitions w/ at least one odd and even part, ranks A366532.
%Y A366527 A086543 counts partitions of n not containing n/2, ranks A366319.
%Y A366527 A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
%Y A366527 Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.
%K A366527 nonn
%O A366527 0,3
%A A366527 _Gus Wiseman_, Oct 16 2023