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

0,3

Comments

Also partitions of 2n with even product.

Links

Table of n, a(n) for n=0..38.

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.

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

Sequence in context: A179904 A376334 A298311 * A161810 A318604 A084631

Adjacent sequences: A366524 A366525 A366526 * A366528 A366529 A366530

Keyword

nonn

Author

Gus Wiseman, Oct 16 2023

Status

approved