A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

0, 1, 1, 2, 1, 2, 3, 2, 1, 5, 5, 2, 5, 2, 8, 18, 1, 2, 19, 2, 24, 41, 20, 2, 9, 44, 31, 94, 102, 2, 125, 2, 1, 206, 68, 365, 382, 2, 98, 433, 155, 2, 716, 2, 1162, 2332, 196, 2, 17, 1108, 563, 1665, 3287, 2, 3906, 5474, 2005, 3083, 509, 2, 9029

Offset

0,4

Comments

The length and median of such a partition are integers with product n.

Links

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

Formula

a(p) = 2 for prime p. - Andrew Howroyd, Jan 21 2023

Example

The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)  (9)
            (111)       (11111)  (222)  (1111111)       (333)
                                 (321)                  (432)
                                                        (531)
                                                        (111111111)
The a(15) = 18 partitions:
  (15)
  (5,5,5)
  (6,5,4)
  (7,5,3)
  (8,5,2)
  (9,5,1)
  (3,3,3,3,3)
  (4,3,3,3,2)
  (4,4,3,2,2)
  (4,4,3,3,1)
  (5,3,3,2,2)
  (5,3,3,3,1)
  (5,4,3,2,1)
  (5,5,3,1,1)
  (6,3,3,2,1)
  (6,4,3,1,1)
  (7,3,3,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]==Median[#]&]], {n, 0, 30}]

Prog

(PARI) \\ P(n, k, m) is g.f. for k parts of max size m.
P(n, k, m)={polcoef(1/prod(i=1, m, 1 - y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ Andrew Howroyd, Jan 21 2023

Crossrefs

This is the odd-length case of A240219, complement A359894, strict A359897.

These partitions are ranked by A359891, complement A359892.

The complement is counted by A359896.

The strict case is A359899, complement A359900.

The version for factorizations is A359910.

A000041 counts partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.

A027193 counts odd-length partitions, strict A067659, ranked by A026424.

A067538 counts ptns with integer mean, strict A102627, ranked by A316413.

A237984 counts ptns containing their mean, strict A240850, ranked by A327473.

A325347 counts ptns with integer median, strict A359907, ranked by A359908.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A008289, A316313, A327472, A327475, A327482, A359889, A359906.

Sequence in context: A352696 A171565 A328266 * A115116 A141662 A328383

Adjacent sequences: A359892 A359893 A359894 * A359896 A359897 A359898

Keyword

nonn

Author

Gus Wiseman, Jan 20 2023

Status

approved