A361656 Number of odd-length integer partitions of n with integer mean.

0, 1, 1, 2, 1, 2, 4, 2, 1, 9, 8, 2, 13, 2, 16, 51, 1, 2, 58, 2, 85, 144, 57, 2, 49, 194, 102, 381, 437, 2, 629, 2, 1, 956, 298, 2043, 1954, 2, 491, 2293, 1116, 2, 4479, 2, 6752, 14671, 1256, 2, 193, 8035, 4570, 11614, 22143, 2, 28585, 39810, 16476, 24691, 4566

Offset: 0

Author

Gus Wiseman, Mar 24 2023

Keywords

• nonn

Comments

These are partitions of n whose length is an odd divisor of n.

Examples

			The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1   2   3     4   5       6     7         8   9           A
          111       11111   222   1111111       333         22222
                            321                 432         32221
                            411                 441         33211
                                                522         42211
                                                531         43111
                                                621         52111
                                                711         61111
                                                111111111
For example, the partition (3,3,2,1,1) has length 5 and mean 2, so is counted under a(10).
		

Links

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

Crossrefs

Odd-length partitions are counted by A027193, bisection A236559.

Including even-length gives A067538 bisected, strict A102627, ranks A316413.

The even-length version is counted by A361655.

A000041 counts integer partitions, strict A000009.

A027187 counts even-length partitions, bisection A236913.

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, A067659, A082550, A237984, A240219, A327475, A348551, A361653.

Programs

Mathematica

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

PARI

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