|
|
A360675
|
|
Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.
|
|
27
|
|
|
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 3, 3, 0, 0, 0, 1, 3, 5, 2, 0, 0, 0, 1, 4, 6, 4, 0, 0, 0, 0, 1, 4, 9, 5, 3, 0, 0, 0, 0, 1, 5, 10, 10, 4, 0, 0, 0, 0, 0, 1, 5, 13, 12, 9, 2, 0, 0, 0, 0, 0, 1, 6, 15, 18, 11, 5, 0, 0, 0, 0, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,8
|
|
COMMENTS
|
Also the number of integer partitions of n whose left half (inclusive) sums to n-k.
|
|
LINKS
|
|
|
EXAMPLE
|
Triangle begins:
1
1 0
1 1 0
1 2 0 0
1 2 2 0 0
1 3 3 0 0 0
1 3 5 2 0 0 0
1 4 6 4 0 0 0 0
1 4 9 5 3 0 0 0 0
1 5 10 10 4 0 0 0 0 0
1 5 13 12 9 2 0 0 0 0 0
1 6 15 18 11 5 0 0 0 0 0 0
1 6 18 22 20 6 4 0 0 0 0 0 0
1 7 20 29 26 13 5 0 0 0 0 0 0 0
1 7 24 34 37 19 11 2 0 0 0 0 0 0 0
1 8 26 44 46 30 16 5 0 0 0 0 0 0 0 0
1 8 30 50 63 40 27 8 4 0 0 0 0 0 0 0 0
1 9 33 61 75 61 36 15 6 0 0 0 0 0 0 0 0 0
1 9 37 70 96 75 61 21 12 3 0 0 0 0 0 0 0 0 0
For example, row n = 9 counts the following partitions:
(9) (81) (72) (63) (54)
(441) (432) (333) (3222)
(531) (522) (3321) (21111111)
(621) (4311) (4221) (111111111)
(711) (5211) (22221)
(6111) (222111)
(32211) (321111)
(33111) (411111)
(42111) (2211111)
(51111) (3111111)
For example, the partition y = (3,2,2,1,1) has right half (exclusive) (1,1), with sum 2, so y is counted under T(9,2).
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], Total[Take[#, -Floor[Length[#]/2]]]==k&]], {n, 0, 18}, {k, 0, n}]
|
|
CROSSREFS
|
The central diagonal T(2n,n) is A000005.
The right inclusive version is A360672 with rows reversed.
A008284 counts partitions by length.
First for prime indices, second for partitions, third for prime factors:
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|