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A361391
Number of strict integer partitions of n with non-integer mean.
2
1, 0, 0, 1, 0, 2, 0, 4, 2, 4, 5, 11, 0, 17, 15, 13, 15, 37, 18, 53, 24, 48, 78, 103, 23, 111, 152, 143, 123, 255, 110, 339, 238, 372, 495, 377, 243, 759, 845, 873, 414, 1259, 842, 1609, 1383, 1225, 2281, 2589, 1285, 2827, 2518, 3904, 3836, 5119, 3715, 4630
OFFSET
0,6
COMMENTS
Are 1, 2, 4, 6, 12 the only zeros?
EXAMPLE
The a(3) = 1 through a(11) = 11 partitions:
{2,1} . {3,2} . {4,3} {4,3,1} {5,4} {5,3,2} {6,5}
{4,1} {5,2} {5,2,1} {6,3} {5,4,1} {7,4}
{6,1} {7,2} {6,3,1} {8,3}
{4,2,1} {8,1} {7,2,1} {9,2}
{4,3,2,1} {10,1}
{5,4,2}
{6,3,2}
{6,4,1}
{7,3,1}
{8,2,1}
{5,3,2,1}
MAPLE
a:= proc(m) option remember; local b; b:=
proc(n, i, t) option remember; `if`(i*(i+1)/2<n,
0, `if`(n=0, signum(irem(m, t)),
b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
end: `if`(m=0, 1, b(m$2, 0))
end:
seq(a(n), n=0..55); # Alois P. Heinz, Mar 16 2023
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!IntegerQ[Mean[#]]&]], {n, 0, 30}]
CROSSREFS
The strict complement is counted by A102627.
The non-strict version is ranked by A348551, complement A316413.
The non-strict version is counted by A349156, complement A067538.
For median instead of mean we have A360952, complement A359907.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A307683 counts partitions with non-integer median, ranks A359912.
A325347 counts partitions with integer median, ranks A359908.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement of A237984.
A327475 counts subsets with integer mean.
Sequence in context: A332001 A196606 A371130 * A337697 A328599 A222303
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 11 2023
EXTENSIONS
a(31)-a(55) from Alois P. Heinz, Mar 16 2023
STATUS
approved