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A361656
Number of odd-length integer partitions of n with integer mean.
2
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
0,4
COMMENTS
These are partitions of n whose length is an odd divisor of n.
LINKS
EXAMPLE
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).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[Mean[#]]&]], {n, 0, 30}]
PROG
(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
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.
Sequence in context: A281729 A302290 A145173 * A270594 A270706 A082793
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 24 2023
STATUS
approved