A343489 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=1..n} k^(gcd(j, n) - 1).

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 2, 0, 1, 4, 6, 4, 4, 0, 1, 5, 11, 12, 5, 2, 0, 1, 6, 18, 32, 20, 6, 6, 0, 1, 7, 27, 70, 85, 42, 7, 4, 0, 1, 8, 38, 132, 260, 260, 70, 8, 6, 0, 1, 9, 51, 224, 629, 1050, 735, 144, 9, 4, 0, 1, 10, 66, 352, 1300, 3162, 4102, 2224, 270, 10, 10

Offset: 0

Author

Seiichi Manyama, Apr 17 2021

Keywords

• nonn
• tabl

Examples

			Square array begins:
  0, 0,  0,   0,    0,    0,    0, ...
  1, 1,  1,   1,    1,    1,    1, ...
  1, 2,  3,   4,    5,    6,    7, ...
  2, 3,  6,  11,   18,   27,   38, ...
  2, 4, 12,  32,   70,  132,  224, ...
  4, 5, 20,  85,  260,  629, 1300, ...
  2, 6, 42, 260, 1050, 3162, 7826, ...
		

Crossrefs

Columns k=0..5 give A000010, A001477, A034738, A034754, A343490, A343492.

Main diagonal gives A056665.

Cf. A185651, A319082.

Programs

Mathematica

T[n_, k_] := Sum[If[k == (g = GCD[j, n] - 1) == 0, 1, k^g], {j, 1, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 17 2021 *)

PARI

T(n, k) = sum(j=1, n, k^(gcd(j, n)-1));

PARI

T(n, k) = if(n==0, 0, sumdiv(n, d, eulerphi(n/d)*k^(d-1)));

Formula

G.f. of column k: Sum_{j>=1} phi(j) * x^j / (1 - k*x^j).

T(n,k) = A185651(n,k)/k for k > 0.

T(n,k) = Sum_{d|n} phi(n/d)*k^(d - 1).