A118229 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).

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, 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, 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

Offset

1,31

Comments

Column 1 is A096433. Column 2 = [0,1,0,-1,0,0,0,...(zero for n>4)]. Column 3 is A118230.

Links

Table of n, a(n) for n=1..105.

Hamed Mousavi, Maxie D. Schmidt, Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums, arXiv:1810.08373 [math.NT], 2018. See Figure 2.1, p. 6.

Formula

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.

Example

Describes a sequence transformation as follows.
Say we have the arbitrary sequence {a(k)}.
We define {b(k)}, based on {a(k)}, by:
b(n) = Sum_{1<=k<=n, gcd(k,n)=1} a(k).
So given {b(k)} (which must have b(1) = b(2)), how do we get the sequence {a(k)}?
If a(n) = Sum_{k >= 2} b(k) * T(n,k), then there is a triangular array {T(n,k)} which begins as follows:
   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,  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,  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; ...

Mathematica

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[n_, k_] := If[n<k || k<0, 0, M[n][[n, k]]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 27 2018, from PARI *)

Prog

(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])

Crossrefs

Cf. A054431 (matrix inverse), A096433 (column 1), A118230 (column 3).

Sequence in context: A321917 A115201 A354100 * A172250 A309047 A255317

Adjacent sequences: A118226 A118227 A118228 * A118230 A118231 A118232

Keyword

sign,tabl

Author

Leroy Quet, Paul D. Hanna, Apr 16 2006

Status

approved