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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
OFFSET
0,8
COMMENTS
Also the number of integer partitions of n whose left half (inclusive) sums to n-k.
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.
Row sums are A000041.
Diagonal sums are A360671, exclusive A360673.
The right inclusive version is A360672 with rows reversed.
The left version has central diagonal A360674, ranks A360953.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Sequence in context: A333365 A303065 A325406 * A257900 A362426 A039971
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 27 2023
STATUS
approved