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
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 27 2023
STATUS
approved