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

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

Author

Gus Wiseman, Oct 16 2023

Keywords

• nonn

Comments

Also partitions of 2n with even product.

Examples

			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)
		

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.

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

Programs

Mathematica

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

Formula

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