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Inverse Moebius transform of signed A007318.
6

%I #19 Jun 22 2024 22:38:44

%S 1,0,1,2,-2,1,-1,4,-3,1,2,-4,6,-4,1,0,4,-9,10,-5,1,2,-6,15,-20,15,-6,

%T 1,-2,11,-24,36,-35,21,-7,1,3,-10,29,-56,70,-56,28,-8,1,0,6,-30,80,

%U -125,126,-84,36,-9,1,2,-10,45,-120,210,-252,210,-120,45,-10,1,-2,18,-67,176,-335,463,-462,330,-165,55,-11,1

%N Inverse Moebius transform of signed A007318.

%C A128316 = A000012 * A128315.

%H G. C. Greubel, <a href="/A128315/b128315.txt">Rows n = 1..100 of the triangle, flattened</a>

%F T(n, k) = A051731(n, k) * A130595(n-1, k-1) as infinite lower triangular matrices.

%F T(n, 1) = A048272(n).

%F Sum_{k=1..n} T(n, k) = A000012(n) = 1 (row sums).

%F From _G. C. Greubel_, Jun 22 2024: (Start)

%F T(n, k) = (-1)^k * Sum_{d|n} (-1)^d * binomial(d-1, k-1).

%F T(n, 2) = A325940(n), n >= 2.

%F T(n, 3) = A363615(n), n >= 3.

%F T(n, 4) = A363616(n), n >= 4.

%F T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.

%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A081295(n).

%F Sum_{k=1..n} k*T(n, k) = A000034(n-1), n >= 1.

%F Sum_{k=1..n} (k+1)*T(n, k) = A010693(n-1), n >= 1. (End)

%e First few rows of the triangle:

%e 1;

%e 0, 1;

%e 2, -2, 1;

%e -1, 4, -3, 1;

%e 2, -4, 6, -4, 1;

%e 0, 4, -9, 10, -5, 1;

%e ...

%t A128315[n_, k_]:= (-1)^k*DivisorSum[n, (-1)^#*Binomial[#-1, k-1] &];

%t Table[A128315[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 22 2024 *)

%o (Magma)

%o A128315:= func< n,k | (&+[0^(n mod j)*(-1)^(k+j)*Binomial(j-1, k-1): j in [k..n]]) >;

%o [A128315(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jun 22 2024

%o (SageMath)

%o def A128315(n,k): return sum( 0^(n%j)*(-1)^(k+j)*binomial(j-1,k-1) for j in range(k,n+1))

%o flatten([[A128315(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jun 22 2024

%Y Cf. A000034, A000984, A007318, A010693, A048272, A051731, A128316.

%Y Cf. A130595, A325940, A363615, A363616.

%Y Cf. A000012 (row sums).

%K tabl,sign

%O 1,4

%A _Gary W. Adamson_, Feb 25 2007

%E a(43) = 28 inserted and more terms from _Georg Fischer_, Jun 05 2023