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 A344824 #24 Jun 28 2024 05:05:38
%S A344824 1,1,2,1,1,3,1,0,3,4,1,-1,5,2,5,1,-2,9,-4,4,6,1,-3,15,-20,13,4,7,1,-4,
%T A344824 23,-52,62,-16,6,8,1,-5,33,-106,205,-174,49,4,9,1,-6,45,-188,520,-806,
%U A344824 556,-88,7,10,1,-7,59,-304,1109,-2584,3291,-1660,173,7,11
%N A344824 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{j=1..n} floor(n/j) * (-k)^(j-1).
%H A344824 G. C. Greubel, <a href="/A344824/b344824.txt">Antidiagonals n = 1..50, flattened</a>
%F A344824 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 + k*x^j).
%F A344824 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (-k)^(j-1) * x^j/(1 - x^j).
%F A344824 A(n,k) = Sum_{j=1..n} Sum_{d|j} (-k)^(d - 1).
%F A344824 T(n, k) = Sum_{j=1..(k+1)} floor((k+1)/j) * (k-n+1)^(j-1), for n >= 1, 0 <= k <= n-1 (antidiagonal triangle). - _G. C. Greubel_, Jun 27 2024
%e A344824 Square array, A(n, k), begins:
%e A344824 1, 1, 1, 1, 1, 1, 1, ...
%e A344824 2, 1, 0, -1, -2, -3, -4, ...
%e A344824 3, 3, 5, 9, 15, 23, 33, ...
%e A344824 4, 2, -4, -20, -52, -106, -188, ...
%e A344824 5, 4, 13, 62, 205, 520, 1109, ...
%e A344824 6, 4, -16, -174, -806, -2584, -6636, ...
%e A344824 Antidiagonal triangle, T(n, k), begins:
%e A344824 1;
%e A344824 1, 2;
%e A344824 1, 1, 3;
%e A344824 1, 0, 3, 4;
%e A344824 1, -1, 5, 2, 5;
%e A344824 1, -2, 9, -4, 4, 6;
%e A344824 1, -3, 15, -20, 13, 4, 7;
%e A344824 1, -4, 23, -52, 62, -16, 6, 8;
%e A344824 1, -5, 33, -106, 205, -174, 49, 4, 9;
%e A344824 1, -6, 45, -188, 520, -806, 556, -88, 7, 10;
%t A344824 A[n_, k_] := Sum[If[k == 0 && j == 1, 1, (-k)^(j - 1)] * Quotient[n, j], {j, 1, n}]; Table[A[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 29 2021 *)
%o A344824 (PARI) A(n, k) = sum(j=1, n, n\j*(-k)^(j-1));
%o A344824 (PARI) A(n, k) = sum(j=1, n, sumdiv(j, d, (-k)^(d-1)));
%o A344824 (Magma)
%o A344824 A:= func< n,k | k eq n select n else (&+[Floor(n/j)*(-k)^(j-1): j in [1..n]]) >;
%o A344824 A344824:= func< n,k | A(k+1, n-k-1) >;
%o A344824 [A344824(n,k): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Jun 27 2024
%o A344824 (SageMath)
%o A344824 def A(n,k): return n if k==n else sum((n//j)*(-k)^(j-1) for j in range(1,n+1))
%o A344824 def A344824(n,k): return A(k+1, n-k-1)
%o A344824 flatten([[A344824(n,k) for k in range(n)] for n in range(1,13)]) # _G. C. Greubel_, Jun 27 2024
%Y A344824 Columns k=0..4 give A000027, A059851, A344817, A344818, A344819.
%Y A344824 A(n,n) gives A344820.
%Y A344824 Cf. A344821.
%K A344824 sign,tabl
%O A344824 1,3
%A A344824 _Seiichi Manyama_, May 29 2021