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 A128316 #17 Jun 23 2024 22:02:25
%S A128316 1,1,1,3,-1,1,2,3,-2,1,4,-1,4,-3,1,4,3,-5,7,-4,1,6,-3,10,-13,11,-5,1,
%T A128316 4,8,-14,23,-24,16,-6,1,7,-2,15,-33,46,-40,22,-7,1,7,4,-15,47,-79,86,
%U A128316 -62,29,-8,1,9,-6,30,-73,131,-166,148,-91,37,-9,1,7,12,-37,103,-204,297,-314,239,-128,46,-10,1
%N A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.
%C A128316 A128316 * [1,2,3...] = A000034: [1,2,1,2,...].
%H A128316 G. C. Greubel, <a href="/A128316/b128316.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A128316 Sum_{k=1..n} T(n, k) = A000027(n) (row sums).
%F A128316 T(n, 1) = A059851(n).
%F A128316 From _G. C. Greubel_, Jun 23 2024: (Start)
%F A128316 T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.
%F A128316 T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).
%F A128316 T(2*n-1, n) = (-1)^(n-1)*A026641(n).
%F A128316 T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.
%F A128316 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).
%F A128316 Sum_{k=1..n} k*T(n, k) = A032766(n-1).
%F A128316 Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)
%e A128316 First few rows of the triangle:
%e A128316 1;
%e A128316 1, 1;
%e A128316 3, -1, 1;
%e A128316 2, 3 -2, 1;
%e A128316 4, -1, 4, -3, 1;
%e A128316 4, 3, -5, 7, -4, 1;
%e A128316 6, -3, 10, -13, 11, -5, 1;
%e A128316 4, 8, -14, 23, -24, 16, -6, 1;
%e A128316 ...
%t A128316 T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];
%t A128316 Table[T[n,k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 23 2024 *)
%o A128316 (Magma)
%o A128316 A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;
%o A128316 [A128316(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jun 23 2024
%o A128316 (SageMath)
%o A128316 def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))
%o A128316 flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Jun 23 2024
%Y A128316 Cf. A000012, A000034, A014300, A026641, A059851, A128315.
%Y A128316 Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).
%K A128316 tabl,sign
%O A128316 1,4
%A A128316 _Gary W. Adamson_, Feb 25 2007
%E A128316 a(28) = 1 inserted and more terms from _Georg Fischer_, Jun 06 2023