A382994 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, 2, 0, 3, -2, 3, 4, -6, 12, 0, 5, -12, 33, -16, 5, 6, -20, 72, -84, 40, 0, 7, -30, 135, -264, 255, -60, 7, 8, -42, 228, -640, 1040, -714, 140, 0, 9, -56, 357, -1320, 3145, -4056, 2205, -272, 9, 10, -72, 528, -2436, 7800, -15540, 16408, -6648, 540, 0

Offset: 1

Author

Seiichi Manyama, Apr 12 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,   2,    3,     4,      5,      6,       7, ...
  0,  -2,   -6,   -12,    -20,    -30,     -42, ...
  3,  12,   33,    72,    135,    228,     357, ...
  0, -16,  -84,  -264,   -640,  -1320,   -2436, ...
  5,  40,  255,  1040,   3145,   7800,   16835, ...
  0, -60, -714, -4056, -15540, -46500, -117390, ...
  7, 140, 2205, 16408,  78155, 279972,  823585, ...
		

Crossrefs

Main diagonal gives A382997.

Cf. A000010, A185651, A382993, A382995.

Programs

PARI

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

Formula

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

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