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
Examples
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, ...
Links
- Seiichi Manyama, Antidiagonals n = 1..140, flattened
Crossrefs
Programs
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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 -
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
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.