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

Author

Gus Wiseman, Dec 01 2023

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A268390.

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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

Formula

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