A182616 Number of partitions of 2n that contain odd parts.

0, 1, 3, 8, 17, 35, 66, 120, 209, 355, 585, 946, 1498, 2335, 3583, 5428, 8118, 12013, 17592, 25525, 36711, 52382, 74173, 104303, 145698, 202268, 279153, 383145, 523105, 710655, 960863, 1293314, 1733281, 2313377, 3075425, 4073085, 5374806, 7067863, 9263076

Offset

0,3

Comments

Bisection (even part) of A086543.

Links

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

Formula

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

Example

For n=3 the partitions of 2n are
6 ....................... does not contains odd parts
3 + 3 ................... contains odd parts ........... *
4 + 2 ................... does not contains odd parts
2 + 2 + 2 ............... does not contains odd parts
5 + 1 ................... contains odd parts ........... *
3 + 2 + 1 ............... contains odd parts ........... *
4 + 1 + 1 ............... contains odd parts ........... *
2 + 2 + 1 + 1 ........... contains odd parts ........... *
3 + 1 + 1 + 1 ........... contains odd parts ........... *
2 + 1 + 1 + 1 + 1 ....... contains odd parts ........... *
1 + 1 + 1 + 1 + 1 + 1 ... contains odd parts ........... *
There are 8 partitions of 2n that contain odd parts.
Also p(2n)-p(n) = p(6)-p(3) = 11-3 = 8, where p(n) is the number of partitions of n, so a(3)=8.
From Gus Wiseman, Oct 18 2023: (Start)
For n > 0, also the number of integer partitions of 2n that do not contain n, ranked by A366321. For example, the a(1) = 1 through a(4) = 17 partitions are:
  (2)  (4)     (6)       (8)
       (31)    (42)      (53)
       (1111)  (51)      (62)
               (222)     (71)
               (411)     (332)
               (2211)    (521)
               (21111)   (611)
               (111111)  (2222)
                         (3221)
                         (3311)
                         (5111)
                         (22211)
                         (32111)
                         (221111)
                         (311111)
                         (2111111)
                         (11111111)
(End)

Maple

with(combinat): a:= n-> numbpart(2*n) -numbpart(n): seq(a(n), n=0..35);

Mathematica

Table[Length[Select[IntegerPartitions[2n], n>0&&FreeQ[#, n]&]], {n, 0, 15}] (* Gus Wiseman, Oct 11 2023 *)
Table[Length[Select[IntegerPartitions[2n], Or@@OddQ/@#&]], {n, 0, 15}] (* Gus Wiseman, Oct 11 2023 *)

Crossrefs

Cf. A304710.

Bisection of A086543, with ranks A366322.

The case of all odd parts is A035294, bisection of A000009.

The strict case is A365828.

These partitions have ranks A366530.

A000041 counts integer partitions, strict A000009.

A006477 counts partitions with at least one odd and even part, ranks A366532.

A047967 counts partitions with at least one even part, ranks A324929.

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

A366527 counts partitions of 2n with an even part, ranks A366529.

Cf. A001522, A006827, A058695, A078408, A079122, A231429, A365543, A366321.

Sequence in context: A182734 A327608 A239844 * A159217 A052996 A112523

Adjacent sequences: A182613 A182614 A182615 * A182617 A182618 A182619

Keyword

nonn

Author

Omar E. Pol, Dec 03 2010

Extensions

Edited by Alois P. Heinz, Dec 03 2010

Status

approved