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 A128315 #19 Jun 22 2024 22:38:44
%S A128315 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 A128315 1,-2,11,-24,36,-35,21,-7,1,3,-10,29,-56,70,-56,28,-8,1,0,6,-30,80,
%U A128315 -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 A128315 Inverse Moebius transform of signed A007318.
%C A128315 A128316 = A000012 * A128315.
%H A128315 G. C. Greubel, <a href="/A128315/b128315.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A128315 T(n, k) = A051731(n, k) * A130595(n-1, k-1) as infinite lower triangular matrices.
%F A128315 T(n, 1) = A048272(n).
%F A128315 Sum_{k=1..n} T(n, k) = A000012(n) = 1 (row sums).
%F A128315 From _G. C. Greubel_, Jun 22 2024: (Start)
%F A128315 T(n, k) = (-1)^k * Sum_{d|n} (-1)^d * binomial(d-1, k-1).
%F A128315 T(n, 2) = A325940(n), n >= 2.
%F A128315 T(n, 3) = A363615(n), n >= 3.
%F A128315 T(n, 4) = A363616(n), n >= 4.
%F A128315 T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
%F A128315 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A081295(n).
%F A128315 Sum_{k=1..n} k*T(n, k) = A000034(n-1), n >= 1.
%F A128315 Sum_{k=1..n} (k+1)*T(n, k) = A010693(n-1), n >= 1. (End)
%e A128315 First few rows of the triangle:
%e A128315 1;
%e A128315 0, 1;
%e A128315 2, -2, 1;
%e A128315 -1, 4, -3, 1;
%e A128315 2, -4, 6, -4, 1;
%e A128315 0, 4, -9, 10, -5, 1;
%e A128315 ...
%t A128315 A128315[n_, k_]:= (-1)^k*DivisorSum[n, (-1)^#*Binomial[#-1, k-1] &];
%t A128315 Table[A128315[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 22 2024 *)
%o A128315 (Magma)
%o A128315 A128315:= func< n,k | (&+[0^(n mod j)*(-1)^(k+j)*Binomial(j-1, k-1): j in [k..n]]) >;
%o A128315 [A128315(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jun 22 2024
%o A128315 (SageMath)
%o A128315 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 A128315 flatten([[A128315(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jun 22 2024
%Y A128315 Cf. A000034, A000984, A007318, A010693, A048272, A051731, A128316.
%Y A128315 Cf. A130595, A325940, A363615, A363616.
%Y A128315 Cf. A000012 (row sums).
%K A128315 tabl,sign
%O A128315 1,4
%A A128315 _Gary W. Adamson_, Feb 25 2007
%E A128315 a(43) = 28 inserted and more terms from _Georg Fischer_, Jun 05 2023