A282750 - OEIS

%I #38 Jul 25 2022 05:08:26

%S 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,

%T 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,

%U 2,8,12,12,10,7,5,3,2,1,1,0,6,14,18,18,14

%N 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).

%C Columns 2-10 are A023022-A023030. - _Lars Blomberg_ Mar 08 2017

%C 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

%H Michael De Vlieger, <a href="/A282750/b282750.txt">Table of n, a(n) for n = 1..5050</a> (rows 1 <= n <= 100)

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

%e Triangle begins:

%e    n/k: 1,  2,  3,  4,  5,  6,  7,  8, ...

%e    1:   1;

%e    2:   0,  1;

%e    3:   0,  1,  1;

%e    4:   0,  1,  1,  1;

%e    5:   0,  2,  2,  1,  1;

%e    6:   0,  1,  2,  2,  1,  1;

%e    7:   0,  3,  4,  3,  2,  1,  1;

%e    8:   0,  2,  4,  4,  3,  2,  1,  1;

%e    9:   0,  3,  6,  6,  5,  3,  2,  1,  1;

%e   10:   0,  2,  6,  8,  6,  5,  3,  2,  1,  1;

%e   11:   0,  5, 10, 11, 10,  7,  5,  3,  2,  1,  1;

%e   12:   0,  2,  8, 12, 12, 10,  7,  5,  3,  2,  1,  1;

%e   ...

%e The partitions with their gcd value for n=8, k=2..5:

%e (1, 7)=1, (2, 6)=2, (3, 5)=1, (4, 4)=4, so T(8,2)=2.

%e (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.

%e (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.

%e (1, 1, 1, 1, 4)=1, (1, 1, 1, 2, 3)=1, (1, 1, 2, 2, 2)=1, so T(8,4)=3.

%e (1, 1, 1, 1, 1, 3)=1, (1, 1, 1, 1, 2, 2)=1, so T(8,5)=2.

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

%o (Sage) # uses[DivisorTriangle from A327029, A008284]

%o DivisorTriangle(moebius, A008284, 13) # _Peter Luschny_, Aug 24 2019

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

%Y Row sums = A000837. See also A051424.

%Y For ordinary partition table see A008284.

%K nonn,tabl

%O 1,12

%A _N. J. A. Sloane_, Mar 05 2017

%E Corrected a(30)-a(32) and more terms from _Lars Blomberg_, Mar 08 2017