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%I #22 Jul 28 2019 16:47:11
%S 1,-1,1,-1,0,1,1,-1,-1,1,-1,0,0,0,1,1,0,0,-1,-1,1,1,0,-1,0,-1,0,1,-1,
%T 0,2,-1,0,0,-1,1,-1,0,0,0,1,0,-1,0,1,1,0,-1,1,0,-1,1,-1,-1,1,-1,0,1,0,
%U 0,0,-1,0,0,0,1,1,0,-1,0,0,0,1,0,0,-1,-1,1,3,0,-2,0,-2,0,2,0,-1,0,-1,0,1,-3,0,1,0,3,0,-1,-1,1,0,0,0,-1,1
%N Triangle, read by rows, equal to the matrix inverse of triangle A054431; the inverse transformation that obtains {a(n)} from b(n) = Sum_{1<=k<=n, gcd(k,n)=1} a(k).
%C Column 1 is A096433. Column 2 = [0,1,0,-1,0,0,0,...(zero for n>4)]. Column 3 is A118230.
%H Hamed Mousavi, Maxie D. Schmidt, <a href="https://arxiv.org/abs/1810.08373">Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums</a>, arXiv:1810.08373 [math.NT], 2018. See Figure 2.1, p. 6.
%F For column k > 1: Sum_{i = 2..n, gcd(n,i) = 1} T(i,k) = 1 when n = k+1, 0 elsewhere; for column k = 1: Sum_{i = 2..n, gcd(n,i) = 1} T(i,1) = 1 when n = 1 or 2, 0 elsewhere.
%e Describes a sequence transformation as follows.
%e Say we have the arbitrary sequence {a(k)}.
%e We define {b(k)}, based on {a(k)}, by:
%e b(n) = Sum_{1<=k<=n, gcd(k,n)=1} a(k).
%e So given {b(k)} (which must have b(1) = b(2)), how do we get the sequence {a(k)}?
%e If a(n) = Sum_{k >= 2} b(k) * T(n,k), then there is a triangular array {T(n,k)} which begins as follows:
%e 1;
%e -1, 1;
%e -1, 0, 1;
%e 1, -1, -1, 1;
%e -1, 0, 0, 0, 1;
%e 1, 0, 0, -1, -1, 1;
%e 1, 0, -1, 0, -1, 0, 1;
%e -1, 0, 2, -1, 0, 0, -1, 1;
%e -1, 0, 0, 0, 1, 0, -1, 0, 1;
%e 1, 0, -1, 1, 0, -1, 1, -1, -1, 1;
%e -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1;
%e 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, -1, 1;
%e 3, 0, -2, 0, -2, 0, 2, 0, -1, 0, -1, 0, 1;
%e -3, 0, 1, 0, 3, 0, -1, -1, 1, 0, 0, 0, -1, 1; ...
%t M[n_] := M[n] = Inverse[Table[If[r >= c, If[GCD[r-c+1, c] == 1, 1, 0], 0], {r, 1, n}, {c, 1, n}]];
%t T[n_, k_] := If[n<k || k<0, 0, M[n][[n, k]]];
%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 27 2018, from PARI *)
%o (PARI) T(n,k)=if(n<k || k<0,0,(matrix(n,n,r,c,if(r>=c,if(gcd(r-c+1,c)==1,1,0)))^-1)[n,k])
%Y Cf. A054431 (matrix inverse), A096433 (column 1), A118230 (column 3).
%K sign,tabl
%O 1,31
%A _Leroy Quet_, _Paul D. Hanna_, Apr 16 2006
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