This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A282750 #38 Jul 25 2022 05:08:26
%S A282750 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 A282750 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 A282750 2,8,12,12,10,7,5,3,2,1,1,0,6,14,18,18,14
%N 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).
%C A282750 Columns 2-10 are A023022-A023030. - _Lars Blomberg_ Mar 08 2017
%C A282750 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 A282750 Michael De Vlieger, <a href="/A282750/b282750.txt">Table of n, a(n) for n = 1..5050</a> (rows 1 <= n <= 100)
%F A282750 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 A282750 Triangle begins:
%e A282750 n/k: 1, 2, 3, 4, 5, 6, 7, 8, ...
%e A282750 1: 1;
%e A282750 2: 0, 1;
%e A282750 3: 0, 1, 1;
%e A282750 4: 0, 1, 1, 1;
%e A282750 5: 0, 2, 2, 1, 1;
%e A282750 6: 0, 1, 2, 2, 1, 1;
%e A282750 7: 0, 3, 4, 3, 2, 1, 1;
%e A282750 8: 0, 2, 4, 4, 3, 2, 1, 1;
%e A282750 9: 0, 3, 6, 6, 5, 3, 2, 1, 1;
%e A282750 10: 0, 2, 6, 8, 6, 5, 3, 2, 1, 1;
%e A282750 11: 0, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1;
%e A282750 12: 0, 2, 8, 12, 12, 10, 7, 5, 3, 2, 1, 1;
%e A282750 ...
%e A282750 The partitions with their gcd value for n=8, k=2..5:
%e A282750 (1, 7)=1, (2, 6)=2, (3, 5)=1, (4, 4)=4, so T(8,2)=2.
%e A282750 (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 A282750 (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 A282750 (1, 1, 1, 1, 4)=1, (1, 1, 1, 2, 3)=1, (1, 1, 2, 2, 2)=1, so T(8,4)=3.
%e A282750 (1, 1, 1, 1, 1, 3)=1, (1, 1, 1, 1, 2, 2)=1, so T(8,5)=2.
%t A282750 Table[Length@ Select[IntegerPartitions[n, {k}], GCD @@ # == 1 &], {n, 13}, {k, n}] // Flatten (* _Michael De Vlieger_, Mar 08 2017 *)
%o A282750 (Sage) # uses[DivisorTriangle from A327029, A008284]
%o A282750 DivisorTriangle(moebius, A008284, 13) # _Peter Luschny_, Aug 24 2019
%Y A282750 Cf. A023022-A023030, A101391 (analog for compositions), A282749 (triangle of partitions into pairwise relatively prime parts).
%Y A282750 Row sums = A000837. See also A051424.
%Y A282750 For ordinary partition table see A008284.
%K A282750 nonn,tabl
%O A282750 1,12
%A A282750 _N. J. A. Sloane_, Mar 05 2017
%E A282750 Corrected a(30)-a(32) and more terms from _Lars Blomberg_, Mar 08 2017