OFFSET
0,4
COMMENTS
Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
EXAMPLE
The a(1) = 1 through a(7) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7)
(21) (31) (32) (42) (43)
(111) (41) (51) (52)
(221) (222) (61)
(311) (411) (322)
(2111) (2211) (331)
(11111) (421)
(511)
(2221)
(3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[Accumulate[#], n/2]&]], {n, 0, 15}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2023
STATUS
approved