A347227 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} mu(d)*mu(n/d)*d^k.

1, 1, -2, 1, -3, -2, 1, -5, -4, 1, 1, -9, -10, 2, -2, 1, -17, -28, 4, -6, 4, 1, -33, -82, 8, -26, 12, -2, 1, -65, -244, 16, -126, 50, -8, 0, 1, -129, -730, 32, -626, 252, -50, 0, 1, 1, -257, -2188, 64, -3126, 1394, -344, 0, 3, 4, 1, -513, -6562, 128, -15626, 8052, -2402, 0, 9, 18, -2

Offset: 1

Author

Seiichi Manyama, Aug 24 2021

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
  -2, -3,  -5,   -9,  -17,   -33, ...
  -2, -4, -10,  -28,  -82,  -244, ...
   1,  2,   4,    8,   16,    32, ...
  -2, -6, -26, -126, -626, -3126, ...
   4, 12,  50,  252, 1394,  8052, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..140, flattened

Crossrefs

Columns k=0..5 give A007427, A046692, A053822, A053825, A053826, A178448.

T(n,n) gives A347251.

Programs

Mathematica

T[n_, k_] := DivisorSum[n, MoebiusMu[#] * MoebiusMu[n/#] * #^k &]; Table[T[n - k + 1, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Aug 24 2021 *)

PARI

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

Formula

Dirichlet g.f. of column k: 1/(zeta(s)*zeta(s-k)).