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Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (j * floor(n/j))^k.
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%I #17 Jul 31 2022 13:17:35

%S 1,1,2,1,4,3,1,8,8,4,1,16,22,15,5,1,32,62,57,21,6,1,64,178,219,91,33,

%T 7,1,128,518,849,405,185,41,8,1,256,1522,3315,1843,1053,247,56,9,1,

%U 512,4502,13017,8541,6065,1523,402,69,10,1,1024,13378,51339,40171,35253,9571,2948,545,87,11

%N Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (j * floor(n/j))^k.

%F T(n,k) = Sum_{j=1..n} j^k * Sum_{d|j} (1 - (1 - 1/d)^k) for k > 0.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 4, 8, 16, 32, 64, 128, ...

%e 3, 8, 22, 62, 178, 518, 1522, ...

%e 4, 15, 57, 219, 849, 3315, 13017, ...

%e 5, 21, 91, 405, 1843, 8541, 40171, ...

%e 6, 33, 185, 1053, 6065, 35253, 206345, ...

%e 7, 41, 247, 1523, 9571, 61091, 394987, ...

%t T[n_, k_] := Sum[(j * Floor[n/j])^k, {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jul 31 2022 *)

%o (PARI) T(n, k) = sum(j=1, n, (j*(n\j))^k);

%o (PARI) T(n, k) = if(k==0, n, sum(j=1, n, j^k*sumdiv(j, d, 1-(1-1/d)^k)));

%Y Columns k=0..3 give A001477, A024916, A350123, A356249.

%Y T(n,n) gives A356238.

%Y Cf. A344725.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Jul 31 2022