A282750 Triangle read by rows: T(n,k) is the number of partitions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1, x_2, ..., x_k) = 1 (where 1 <= k <= n).

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 3, 4, 3, 2, 1, 1, 0, 2, 4, 4, 3, 2, 1, 1, 0, 3, 6, 6, 5, 3, 2, 1, 1, 0, 2, 6, 8, 6, 5, 3, 2, 1, 1, 0, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 2, 8, 12, 12, 10, 7, 5, 3, 2, 1, 1, 0, 6, 14, 18, 18, 14

Offset

1,12

Comments

Columns 2-10 are A023022-A023030. - Lars Blomberg Mar 08 2017

To base the triangle on (0, 0) a column (1, 0, 0, ...) has to be appended to the left hand side of the triangle. To compute this triangle with Michael De Vlieger's Mathematica program only the ranges of the indices have to be adapted. The SageMath program computes the extended triangle by default. - Peter Luschny, Aug 24 2019

Links

Michael De Vlieger, Table of n, a(n) for n = 1..5050 (rows 1 <= n <= 100)

Formula

T(n, k) = Sum_{d|n} Moebius(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. - Peter Luschny, Aug 24 2019

Example

Triangle begins:
   n/k: 1,  2,  3,  4,  5,  6,  7,  8, ...
   1:   1;
   2:   0,  1;
   3:   0,  1,  1;
   4:   0,  1,  1,  1;
   5:   0,  2,  2,  1,  1;
   6:   0,  1,  2,  2,  1,  1;
   7:   0,  3,  4,  3,  2,  1,  1;
   8:   0,  2,  4,  4,  3,  2,  1,  1;
   9:   0,  3,  6,  6,  5,  3,  2,  1,  1;
  10:   0,  2,  6,  8,  6,  5,  3,  2,  1,  1;
  11:   0,  5, 10, 11, 10,  7,  5,  3,  2,  1,  1;
  12:   0,  2,  8, 12, 12, 10,  7,  5,  3,  2,  1,  1;
  ...
The partitions with their gcd value for n=8, k=2..5:
(1, 7)=1, (2, 6)=2, (3, 5)=1, (4, 4)=4, so T(8,2)=2.
(1, 1, 6)=1, (1, 2, 5)=1, (1, 3, 4)=1, (2, 2, 4)=2, (2, 3, 3)=1, so T(8,2)=4.
(1, 1, 1, 5)=1, (1, 1, 2, 4)=1, (1, 1, 3, 3)=1, (1, 2, 2, 3)=1, (2, 2, 2, 2)=2, so T(8,3)=4.
(1, 1, 1, 1, 4)=1, (1, 1, 1, 2, 3)=1, (1, 1, 2, 2, 2)=1, so T(8,4)=3.
(1, 1, 1, 1, 1, 3)=1, (1, 1, 1, 1, 2, 2)=1, so T(8,5)=2.

Mathematica

Table[Length@ Select[IntegerPartitions[n, {k}], GCD @@ # == 1 &], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Mar 08 2017 *)

Prog

(Sage) # uses[DivisorTriangle from A327029, A008284]
DivisorTriangle(moebius, A008284, 13) # Peter Luschny, Aug 24 2019

Crossrefs

Cf. A023022-A023030, A101391 (analog for compositions), A282749 (triangle of partitions into pairwise relatively prime parts).

Row sums = A000837. See also A051424.

For ordinary partition table see A008284.

Sequence in context: A255212 A323011 A327747 * A265890 A226920 A362759

Adjacent sequences: A282747 A282748 A282749 * A282751 A282752 A282753

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 05 2017

Extensions

Corrected a(30)-a(32) and more terms from Lars Blomberg, Mar 08 2017

Status

approved