A303138 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A303138

Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

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

Table of n, a(n) for n=1..110.

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.

Sequence in context: A060016 A117408 A228360 * A276205 A244966 A079100

Adjacent sequences: A303135 A303136 A303137 * A303139 A303140 A303141

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Apr 19 2018

STATUS

approved