A322324 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).

1, 1, 0, 1, -1, 0, 1, -3, -2, 0, 1, -7, -8, -1, 0, 1, -15, -26, -3, -4, 0, 1, -31, -80, -7, -24, 2, 0, 1, -63, -242, -15, -124, 24, -6, 0, 1, -127, -728, -31, -624, 182, -48, -1, 0, 1, -255, -2186, -63, -3124, 1200, -342, -3, -2, 0, 1, -511, -6560, -127, -15624, 7502, -2400, -7, -8, 4, 0

Offset

1,8

Links

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

Formula

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

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

A(n,k) = Sum_{d|n} mu(d)*d^k.

Example

Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -2,  -8,  -26,   -80,  -242, ...
  0, -1,  -3,   -7,   -15,   -31, ...
  0, -4, -24, -124,  -624, -3124, ...
  0,  2,  24,  182,  1200,  7502, ...

Mathematica

Table[Function[k, Product[1 - Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j] j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[d] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

Prog

(PARI) T(n, k) = sumdiv(n, d, moebius(d)*d^k);
matrix(6, 6, n, k, T(n, k-1)) \\ Michel Marcus, Dec 03 2018

Crossrefs

Columns k=0..5 give A063524, A023900, A046970, A063453, A189922, A189923.

Cf. A008683, A059379, A059380, A321222 (diagonal).

Sequence in context: A253669 A154477 A363674 * A142071 A350448 A291680

Adjacent sequences: A322321 A322322 A322323 * A322325 A322326 A322327

Keyword

sign,tabl

Author

Ilya Gutkovskiy, Dec 03 2018

Status

approved