A346148 Square array read by descending antidiagonals: T(n, k) = mu^n(k) where mu^1(k) = mu(k) = A008683(k) and for each n >= 1, mu^(n+1)(k) is the Dirichlet convolution of mu(k) and mu^n(k).
1, -1, 1, -1, -2, 1, 0, -2, -3, 1, -1, 1, -3, -4, 1, 1, -2, 3, -4, -5, 1, -1, 4, -3, 6, -5, -6, 1, 0, -2, 9, -4, 10, -6, -7, 1, 0, 0, -3, 16, -5, 15, -7, -8, 1, 1, 1, -1, -4, 25, -6, 21, -8, -9, 1, -1, 4, 3, -4, -5, 36, -7, 28, -9, -10, 1, 0, -2, 9, 6, -10, -6
Offset: 1
Examples
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 ... ---+-------------------------------------------------------------- 1 | 1 -1 -1 0 -1 1 -1 0 0 1 -1 0 ... 2 | 1 -2 -2 1 -2 4 -2 0 1 4 -2 -2 ... 3 | 1 -3 -3 3 -3 9 -3 -1 3 9 -3 -9 ... 4 | 1 -4 -4 6 -4 16 -4 -4 6 16 -4 -24 ... 5 | 1 -5 -5 10 -5 25 -5 -10 10 25 -5 -50 ... 6 | 1 -6 -6 15 -6 36 -6 -20 15 36 -6 -90 ... 7 | 1 -7 -7 21 -7 49 -7 -35 21 49 -7 -147 ... 8 | 1 -8 -8 28 -8 64 -8 -56 28 64 -8 -224 ... 9 | 1 -9 -9 36 -9 81 -9 -84 36 81 -9 -324 ... 10 | 1 -10 -10 45 -10 100 -10 -120 45 100 -10 -450 ... 11 | 1 -11 -11 55 -11 121 -11 -165 55 121 -11 -605 ... 12 | 1 -12 -12 66 -12 144 -12 -220 66 144 -12 -792 ... 13 | 1 -13 -13 78 -13 169 -13 -286 78 169 -13 -1014 ... 14 | 1 -14 -14 91 -14 196 -14 -364 91 196 -14 -1274 ... 15 | 1 -15 -15 105 -15 225 -15 -455 105 225 -15 -1575 ... ...
Links
- Sebastian Karlsson, Antidiagonals n = 1..140, flattened
Crossrefs
Programs
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Mathematica
T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 13 2021 *) -
PARI
T(n, k) = my(f=factor(k)); for (k=1, #f~, f[k,1] = binomial(n, f[k,2])*(-1)^f[k,2]; f[k,2]=1); factorback(f); \\ Michel Marcus, Aug 21 2021
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Python
from sympy import binomial, primefactors as pf, multiplicity as mult from math import prod def T(n, k): return prod((-1)**mult(p, k)*binomial(n, mult(p, k)) for p in pf(k))
Formula
If k = Product (p_j^m_j) then T(n, k) = Product (binomial(n, m_j)*(-1)^m_j).
Dirichlet g.f. of the n-th row: 1/zeta^n(s).
T(n, p) = -n.
T(n, n) = A341837(n).