This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346148 #40 Sep 13 2021 11:38:05
%S A346148 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,
%T A346148 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,
%U A346148 -9,1,-1,4,3,-4,-5,36,-7,28,-9,-10,1,0,-2,9,6,-10,-6
%N 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).
%H A346148 Sebastian Karlsson, <a href="/A346148/b346148.txt">Antidiagonals n = 1..140, flattened</a>
%F A346148 If k = Product (p_j^m_j) then T(n, k) = Product (binomial(n, m_j)*(-1)^m_j).
%F A346148 Dirichlet g.f. of the n-th row: 1/zeta^n(s).
%F A346148 T(n, p) = -n.
%F A346148 T(n, n) = A341837(n).
%e A346148 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e A346148 ---+--------------------------------------------------------------
%e A346148 1 | 1 -1 -1 0 -1 1 -1 0 0 1 -1 0 ...
%e A346148 2 | 1 -2 -2 1 -2 4 -2 0 1 4 -2 -2 ...
%e A346148 3 | 1 -3 -3 3 -3 9 -3 -1 3 9 -3 -9 ...
%e A346148 4 | 1 -4 -4 6 -4 16 -4 -4 6 16 -4 -24 ...
%e A346148 5 | 1 -5 -5 10 -5 25 -5 -10 10 25 -5 -50 ...
%e A346148 6 | 1 -6 -6 15 -6 36 -6 -20 15 36 -6 -90 ...
%e A346148 7 | 1 -7 -7 21 -7 49 -7 -35 21 49 -7 -147 ...
%e A346148 8 | 1 -8 -8 28 -8 64 -8 -56 28 64 -8 -224 ...
%e A346148 9 | 1 -9 -9 36 -9 81 -9 -84 36 81 -9 -324 ...
%e A346148 10 | 1 -10 -10 45 -10 100 -10 -120 45 100 -10 -450 ...
%e A346148 11 | 1 -11 -11 55 -11 121 -11 -165 55 121 -11 -605 ...
%e A346148 12 | 1 -12 -12 66 -12 144 -12 -220 66 144 -12 -792 ...
%e A346148 13 | 1 -13 -13 78 -13 169 -13 -286 78 169 -13 -1014 ...
%e A346148 14 | 1 -14 -14 91 -14 196 -14 -364 91 196 -14 -1274 ...
%e A346148 15 | 1 -15 -15 105 -15 225 -15 -455 105 225 -15 -1575 ...
%e A346148 ...
%t A346148 T[n_, k_] := If[k == 1, 1, Product[(-1)^e Binomial[n, e], {e, FactorInteger[k][[All, 2]]}]];
%t A346148 Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Sep 13 2021 *)
%o A346148 (Python)
%o A346148 from sympy import binomial, primefactors as pf, multiplicity as mult
%o A346148 from math import prod
%o A346148 def T(n, k):
%o A346148 return prod((-1)**mult(p, k)*binomial(n, mult(p, k)) for p in pf(k))
%o A346148 (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
%Y A346148 Row n=1..10 give A008683, A007427, A007428, A247343, A341831, A341832, A341833, A341834, A341835, A341836.
%Y A346148 Main diagonal gives A341837.
%Y A346148 Cf. A077592, A163767,
%K A346148 sign,tabl
%O A346148 1,5
%A A346148 _Sebastian Karlsson_, Aug 20 2021