A367588 Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 5, 9, 6, 9, 10, 11, 8, 15, 9, 16, 14, 15, 11, 23, 14, 18, 18, 23, 14, 30, 15, 26, 22, 24, 22, 38, 18, 27, 26, 38, 20, 42, 21, 37, 36, 33, 23, 53, 27, 42, 34, 44, 26, 54, 34, 53, 38, 42, 29, 74, 30, 45, 49, 57, 40, 66, 33, 58, 46

Offset

0,6

Comments

The Heinz numbers of these partitions are given by A268390.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: Sum_{i, j>0} x^(j*(2*i+1))/(1-x^j). - John Tyler Rascoe, Feb 04 2024

Example

The a(3) = 1 through a(12) = 9 partitions (A = 10, B = 11):
  (21)  (31)  (32)  (42)    (43)  (53)    (54)      (64)    (65)  (75)
              (41)  (51)    (52)  (62)    (63)      (73)    (74)  (84)
                    (2211)  (61)  (71)    (72)      (82)    (83)  (93)
                                  (3311)  (81)      (91)    (92)  (A2)
                                          (222111)  (3322)  (A1)  (B1)
                                                    (4411)        (4422)
                                                                  (5511)
                                                                  (333111)
                                                                  (22221111)

Mathematica

Table[Sum[Floor[(d-1)/2], {d, Divisors[n]}], {n, 30}]

Crossrefs

For any multiplicities we have A002133, ranks A007774.

For any number of distinct parts we have A047966, ranks A072774.

For distinct multiplicities we have A182473, ranks A367589.

These partitions have ranks A268390.

A000041 counts integer partitions, strict A000009.

A072233 counts partitions by number of parts.

A116608 counts partitions by number of distinct parts.

Cf. A023645, A091602, A116861, A181819, A243978, A367582.

Sequence in context: A332682 A304183 A157222 * A320033 A333527 A064047

Adjacent sequences: A367585 A367586 A367587 * A367589 A367590 A367591

Keyword

nonn

Author

Gus Wiseman, Dec 01 2023

Status

approved