A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

1, 2, -1, 3, -3, 0, 4, -6, 2, 0, 5, -10, 8, -3, 0, 6, -15, 20, -18, 6, 0, 7, -21, 40, -60, 48, -11, 0, 8, -28, 70, -150, 204, -124, 18, 0, 9, -36, 112, -315, 624, -690, 312, -30, 0, 10, -45, 168, -588, 1554, -2620, 2340, -810, 56, 0, 11, -55, 240, -1008, 3360, -7805, 11160, -8160, 2184, -105, 0

Offset: 1

Author

Seiichi Manyama, Apr 12 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,   2,    3,    4,     5,     6,      7, ...
  -1,  -3,   -6,  -10,   -15,   -21,    -28, ...
   0,   2,    8,   20,    40,    70,    112, ...
   0,  -3,  -18,  -60,  -150,  -315,   -588, ...
   0,   6,   48,  204,   624,  1554,   3360, ...
   0, -11, -124, -690, -2620, -7805, -19656, ...
   0,  18,  312, 2340, 11160, 39990, 117648, ...
		

Crossrefs

Columns k=1..5 give A154955, A038063, A038064, A038065, A038066.

Main diagonal gives A383012.

Cf. A008683, A074650.

Programs

PARI

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

Formula

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

Product_{n>=1} 1/(1 - x^n)^A(n,k) = 1 + k*x.