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 A344821 #26 Jun 28 2024 05:05:30
%S A344821 1,1,2,1,3,3,1,4,5,4,1,5,9,8,5,1,6,15,20,10,6,1,7,23,46,37,14,7,1,8,
%T A344821 33,92,128,76,16,8,1,9,45,164,349,384,141,20,9,1,10,59,268,790,1394,
%U A344821 1114,280,23,10,1,11,75,410,1565,3946,5491,3332,541,27,11
%N A344821 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 A344821 G. C. Greubel, <a href="/A344821/b344821.txt">Antidiagonals n = 1..50, flattened</a>
%F A344821 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - k*x^j).
%F A344821 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} k^(j-1) * x^j/(1 - x^j).
%F A344821 A(n, k) = Sum_{j=1..n} Sum_{d|j} k^(d - 1).
%F A344821 T(n, k) = Sum_{j=1..k+1} floor((k+1)/j) * (n-k-1)^(j-1), for n >= 1, 0 <= k <= n-1 (antidiagonal triangle). - _G. C. Greubel_, Jun 27 2024
%e A344821 Square array, A(n, k), begins:
%e A344821 1, 1, 1, 1, 1, 1, 1, ...
%e A344821 2, 3, 4, 5, 6, 7, 8, ...
%e A344821 3, 5, 9, 15, 23, 33, 45, ...
%e A344821 4, 8, 20, 46, 92, 164, 268, ...
%e A344821 5, 10, 37, 128, 349, 790, 1565, ...
%e A344821 6, 14, 76, 384, 1394, 3946, 9384, ...
%e A344821 Antidiagonal triangle, T(n, k), begins:
%e A344821 1;
%e A344821 1, 2;
%e A344821 1, 3, 3;
%e A344821 1, 4, 5, 4;
%e A344821 1, 5, 9, 8, 5;
%e A344821 1, 6, 15, 20, 10, 6;
%e A344821 1, 7, 23, 46, 37, 14, 7;
%e A344821 1, 8, 33, 92, 128, 76, 16, 8;
%e A344821 1, 9, 45, 164, 349, 384, 141, 20, 9;
%t A344821 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 A344821 (PARI) A(n, k) = sum(j=1, n, n\j*k^(j-1));
%o A344821 (PARI) A(n, k) = sum(j=1, n, sumdiv(j, d, k^(d-1)));
%o A344821 (Magma)
%o A344821 A:= func< n,k | k eq n select n else (&+[Floor(n/j)*k^(j-1): j in [1..n]]) >;
%o A344821 A344821:= func< n,k | A(k+1, n-k-1) >;
%o A344821 [A344821(n,k): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Jun 27 2024
%o A344821 (SageMath)
%o A344821 def A(n,k): return n if k==n else sum((n//j)*k^(j-1) for j in range(1,n+1))
%o A344821 def A344821(n,k): return A(k+1, n-k-1)
%o A344821 flatten([[A344821(n,k) for k in range(n)] for n in range(1,13)]) # _G. C. Greubel_, Jun 27 2024
%Y A344821 Columns k=0..5 give A000027, A006218, A268235, A344814, A344815, A344816.
%Y A344821 A(n,n) gives A332533.
%Y A344821 Cf. A308813, A344824.
%K A344821 nonn,tabl
%O A344821 1,3
%A A344821 _Seiichi Manyama_, May 29 2021