A361655 Number of even-length integer partitions of 2n with integer mean.

0, 1, 3, 4, 10, 6, 33, 8, 65, 68, 117, 12, 583, 14, 319, 1078, 1416, 18, 3341, 20, 8035, 5799, 1657, 24, 36708, 16954, 3496, 24553, 68528, 30, 192180, 32, 178802, 91561, 14625, 485598, 955142, 38, 29223, 316085, 2622697, 42, 3528870, 44, 2443527, 5740043

Offset: 0

Author

Gus Wiseman, Mar 23 2023

Keywords

• nonn

Examples

			The a(0) = 0 through a(5) = 6 partitions:
  .  (11)  (22)    (33)      (44)        (55)
           (31)    (42)      (53)        (64)
           (1111)  (51)      (62)        (73)
                   (111111)  (71)        (82)
                             (2222)      (91)
                             (3221)      (1111111111)
                             (3311)
                             (4211)
                             (5111)
                             (11111111)
For example, the partition (4,2,1,1) has length 4 and mean 2, so is counted under a(4).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Even-length partitions are counted by A027187, bisection A236913.

Including odd-length partitions gives A067538 bisected, ranks A316413.

For median instead of mean we have A361653.

The odd-length version is counted by A361656.

A000041 counts integer partitions, strict A000009.

A051293 counts subsets with integer mean, median A000975.

A058398 counts partitions by mean, see also A008284, A327482.

A325347 counts partitions with integer median, complement A307683.

A326567/A326568 gives mean of prime indices.

A326622 counts factorizations with integer mean.

Cf. A000016, A027193, A067659, A082550, A102627, A237984, A240219, A327475, A348551, A359893.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,15}]

PARI

a(n)=if(n==0, 0, sumdiv(n, d, polcoef(1/prod(k=1, 2*d, 1 - x^k + O(x*x^(2*(n-d)))), 2*(n-d)))) \\ Andrew Howroyd, Mar 24 2023

Extensions

Terms a(36) and beyond from Andrew Howroyd, Mar 24 2023