A382995 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = Sum_{d|n} phi(n/d) * (-k)^(d-1).

1, 1, 0, 1, -1, 3, 1, -2, 6, 0, 1, -3, 11, -8, 5, 1, -4, 18, -28, 20, 0, 1, -5, 27, -66, 85, -30, 7, 1, -6, 38, -128, 260, -238, 70, 0, 1, -7, 51, -220, 629, -1014, 735, -136, 9, 1, -8, 66, -348, 1300, -3108, 4102, -2216, 270, 0, 1, -9, 83, -518, 2405, -7750, 15631, -16452, 6585, -500, 11

Offset: 1

Author

Seiichi Manyama, Apr 12 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   1,    1,     1,     1,     1,      1, ...
  0,  -1,   -2,    -3,    -4,    -5,     -6, ...
  3,   6,   11,    18,    27,    38,     51, ...
  0,  -8,  -28,   -66,  -128,  -220,   -348, ...
  5,  20,   85,   260,   629,  1300,   2405, ...
  0, -30, -238, -1014, -3108, -7750, -16770, ...
  7,  70,  735,  4102, 15631, 46662, 117655, ...
		

Crossrefs

Columns k=1..3 give A193356, A382999, A383000.

Main diagonal gives A382998.

Cf. A000010, A343489, A382994.

Programs

PARI

a(n, k) = sumdiv(n, d, eulerphi(n/d)*(-k)^(d-1));

Formula

A(n,k) = (1/k) * A382994(n,k).

A(n,k) = Sum_{j=1..n} (-k)^(gcd(n,j) - 1).

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