|
|
A365825
|
|
Number of integer partitions of n that are not of length 2 and do not contain n/2.
|
|
5
|
|
|
1, 1, 1, 2, 2, 5, 6, 12, 14, 26, 31, 51, 61, 95, 114, 169, 202, 289, 347, 481, 576, 782, 936, 1244, 1487, 1946, 2323, 2997, 3570, 4551, 5414, 6827, 8103, 10127, 11997, 14866, 17575, 21619, 25507, 31166, 36692, 44563, 52362, 63240, 74152, 89112, 104281, 124731
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
Also the number of integer partitions of n with no two possibly equal parts summing to n.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The a(1) = 1 through a(8) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(111) (1111) (221) (222) (322) (332)
(311) (411) (331) (521)
(2111) (2211) (421) (611)
(11111) (21111) (511) (2222)
(111111) (2221) (3221)
(3211) (3311)
(4111) (5111)
(22111) (22211)
(31111) (32111)
(211111) (221111)
(1111111) (311111)
(2111111)
(11111111)
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], Length[#]!=2&&FreeQ[#, n/2]&]], {n, 0, 15}]
|
|
PROG
|
(Python)
from sympy import npartitions
def A365825(n): return npartitions(n)-(m:=n>>1)-(0 if n&1 else npartitions(m)-1) # Chai Wah Wu, Sep 23 2023
|
|
CROSSREFS
|
The complement is counted by A238628.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A182616 counts partitions of 2n that do not contain n, strict A365828.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|