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

%I A322324 #16 Apr 20 2019 08:07:35
%S A322324 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,
%T A322324 -24,2,0,1,-63,-242,-15,-124,24,-6,0,1,-127,-728,-31,-624,182,-48,-1,
%U A322324 0,1,-255,-2186,-63,-3124,1200,-342,-3,-2,0,1,-511,-6560,-127,-15624,7502,-2400,-7,-8,4,0
%N A322324 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).
%H A322324 Seiichi Manyama, <a href="/A322324/b322324.txt">Antidiagonals n = 1..140, flattened</a>
%F A322324 G.f. of column k: Sum_{j>=1} mu(j)*j^k*x^j/(1 - x^j).
%F A322324 Dirichlet g.f. of column k: zeta(s)/zeta(s-k).
%F A322324 A(n,k) = Sum_{d|n} mu(d)*d^k.
%e A322324 Square array begins:
%e A322324   1,  1,   1,    1,     1,     1, ...
%e A322324   0, -1,  -3,   -7,   -15,   -31, ...
%e A322324   0, -2,  -8,  -26,   -80,  -242, ...
%e A322324   0, -1,  -3,   -7,   -15,   -31, ...
%e A322324   0, -4, -24, -124,  -624, -3124, ...
%e A322324   0,  2,  24,  182,  1200,  7502, ...
%t A322324 Table[Function[k, Product[1 - Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A322324 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
%t A322324 Table[Function[k, Sum[MoebiusMu[d] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%o A322324 (PARI) T(n, k) = sumdiv(n, d, moebius(d)*d^k);
%o A322324 matrix(6, 6, n, k, T(n, k-1)) \\ _Michel Marcus_, Dec 03 2018
%Y A322324 Columns k=0..5 give A063524, A023900, A046970, A063453, A189922, A189923.
%Y A322324 Cf. A008683, A059379, A059380, A321222 (diagonal).
%K A322324 sign,tabl
%O A322324 1,8
%A A322324 _Ilya Gutkovskiy_, Dec 03 2018