|
|
A303138
|
|
Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.
|
|
7
|
|
|
1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 0, 2, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,11
|
|
LINKS
|
|
|
FORMULA
|
If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.
|
|
EXAMPLE
|
Triangle begins:
01: 1
02: 0 1
03: 1 0 1
04: 1 0 0 1
05: 2 0 0 0 1
06: 2 1 0 0 0 1
07: 4 0 0 0 0 0 1
08: 4 1 0 0 0 0 0 1
09: 6 0 1 0 0 0 0 0 1
10: 7 2 0 0 0 0 0 0 0 1
11: 11 0 0 0 0 0 0 0 0 0 1
12: 10 2 1 1 0 0 0 0 0 0 0 1
13: 17 0 0 0 0 0 0 0 0 0 0 0 1
14: 17 4 0 0 0 0 0 0 0 0 0 0 0 1
15: 23 0 2 0 1 0 0 0 0 0 0 0 0 0 1
The strict partitions counted in row 12 are the following.
T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
T(12,2) = 2: (10,2) (6,4,2)
T(12,3) = 1: (9,3)
T(12,4) = 1: (8,4)
T(12,12) = 1: (12)
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&GCD@@#===k&]], {n, 15}, {k, n}]
|
|
CROSSREFS
|
First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.
Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|