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%I #11 Jun 25 2024 08:32:14
%S 1,0,1,3,-2,1,0,4,-3,1,5,-4,6,-4,1,0,5,-9,10,-5,1,7,-6,15,-20,15,-6,1,
%T 0,12,-24,36,-35,21,-7,1,9,-12,30,-56,70,-56,28,-8,1,0,9,-30,80,-125,
%U 126,-84,36,-9,1,11,-10,45,-120,210,-252,210,-120,45,-10,1
%N Triangle read by rows: T = A054523 * A130595, as a lower triangular matrix.
%H G. C. Greubel, <a href="/A128317/b128317.txt">Rows n = 1..100 of the triangle, flattened</a>
%F Equals A054523 * signed A007318 as infinite lower triangular matrices. A007318 is signed by columns: (+, -, +, ...).
%F Sum_{k=1..n} T(n, k) = A000010(n) (row sums).
%F From _G. C. Greubel_, Jun 24 2024: (Start)
%F T(n, k) = A054523 * A130595, as a lower triangular matrix.
%F T(n, k) = Sum_{j=k..n} (-1)^(k+j)*A054523(n,j)*binomial(j-1, k-1).
%F T(n, k) = Sum_{d|n} (-1)^(d+k)*EulerPhi(n/d)*binomial(d-1, k-1).
%F T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
%F T(2*n-2, n-1) = (-1)^n*A001700(n-2), n >= 2.
%F Sum_{k=1..n} k*T(n, k) = A126246(n). (End)
%e First few rows of the triangle:
%e 1;
%e 0, 1;
%e 3, -2, 1;
%e 0, 4, -3, 1;
%e 5, -4, 6, -4, 1;
%e 0, 5, -9, 10, -5, 1;
%e 7, -6, 15, -20, 15, -6, 1;
%e ...
%t A128317[n_, k_]:= DivisorSum[n, (-1)^(#+k)*EulerPhi[n/#]*Binomial[#-1, k-1] &];
%t Table[A128317[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 24 2024 *)
%o (Magma)
%o A128317:= func< n,k | (&+[(-1)^(d+k)*EulerPhi(Floor(n/d))*Binomial(d-1, k-1) : d in Divisors(n)]) >;
%o [A128317(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jun 24 2024
%o (SageMath)
%o def A128317(n,k): return sum((-1)^(k+j)*euler_phi(n/j)*binomial(j-1, k-1) for j in (1..n) if (j).divides(n))
%o flatten([[A128317(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jun 24 2024
%Y Cf. A000984, A001700, A007318, A054523, A130595.
%Y Sums include: A000010 (row sums), A126246.
%K tabl,sign
%O 1,4
%A _Gary W. Adamson_, Feb 25 2007
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