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 A143239 #9 Sep 12 2024 11:34:44
%S A143239 1,2,-1,3,0,-1,4,-2,0,0,5,0,0,0,-1,6,-3,-2,0,0,1,7,0,0,0,0,0,-1,8,-4,
%T A143239 0,0,0,0,0,0,9,0,-3,0,0,0,0,0,0,10,-5,0,0,-2,0,0,0,0,1,11,0,0,0,0,0,0,
%U A143239 0,0,0,-1,12,-6,-4,0,0,2,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,-1,14,-7,0,0,0,0,-2,0,0,0,0,0,0,1
%N A143239 Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.
%C A143239 Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); as a consequence of the Dedekind-Liouville rule illustrated in the example and on p. 137 of "Concrete Mathematics".
%D A143239 Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2nd ed.; Addison-Wesley, 1994, p. 137.
%H A143239 G. C. Greubel, <a href="/A143239/b143239.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A143239 Triangle read by rows generated from the Dedekind-Liouville rule: T(n,k) = mu(k)*(n/k) if k divides n, otherwise T(n,k) = 0 if k is not a divisor of n.
%F A143239 Equals A126988 * A128407.
%e A143239 First few rows of the triangle are:
%e A143239 1;
%e A143239 2, -1;
%e A143239 3, 0, -1;
%e A143239 4, -2, 0, 0;
%e A143239 5, 0, 0, 0, -1;
%e A143239 6, -3, -2, 0, 0, 1;
%e A143239 7, 0, 0, 0, 0, 0, -1;
%e A143239 8, -4, 0, 0, 0, 0, 0, 0;
%e A143239 9, 0, -3, 0, 0, 0, 0, 0, 0;
%e A143239 10, -5, 0, 0, -2, 0, 0, 0, 0, 1;
%e A143239 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%e A143239 12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0;
%e A143239 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%e A143239 14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1;
%e A143239 ...
%e A143239 Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) sum = 4 = phi(12).
%t A143239 A143239[n_, k_]:= If[Mod[n,k]==0, MoebiusMu[k]*(n/k), 0];
%t A143239 Table[A143239[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Sep 12 2024 *)
%o A143239 (Magma)
%o A143239 A143239:= func< n,k | (n mod k) eq 0 select MoebiusMu(k)*(n/k) else 0 >;
%o A143239 [A143239(n,k): k in [1..n], n in [1..14]]; // _G. C. Greubel_, Sep 12 2024
%o A143239 (SageMath)
%o A143239 def A143239(n,k): return moebius(k)*(n//k) if (n%k)==0 else 0
%o A143239 flatten([[A143239(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Sep 12 2024
%Y A143239 Cf. A000010 (row sums), A008683, A126988, A128407.
%K A143239 tabl,sign
%O A143239 1,2
%A A143239 _Gary W. Adamson_, Aug 01 2008