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A356250 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.

%I A356250 #17 Jul 31 2022 13:17:35
%S A356250 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 A356250 7,1,128,518,849,405,185,41,8,1,256,1522,3315,1843,1053,247,56,9,1,
%U A356250 512,4502,13017,8541,6065,1523,402,69,10,1,1024,13378,51339,40171,35253,9571,2948,545,87,11
%N A356250 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 A356250 T(n,k) = Sum_{j=1..n} j^k * Sum_{d|j} (1 - (1 - 1/d)^k) for k > 0.
%e A356250 Square array begins:
%e A356250   1,  1,   1,    1,    1,     1,      1, ...
%e A356250   2,  4,   8,   16,   32,    64,    128, ...
%e A356250   3,  8,  22,   62,  178,   518,   1522, ...
%e A356250   4, 15,  57,  219,  849,  3315,  13017, ...
%e A356250   5, 21,  91,  405, 1843,  8541,  40171, ...
%e A356250   6, 33, 185, 1053, 6065, 35253, 206345, ...
%e A356250   7, 41, 247, 1523, 9571, 61091, 394987, ...
%t A356250 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 A356250 (PARI) T(n, k) = sum(j=1, n, (j*(n\j))^k);
%o A356250 (PARI) T(n, k) = if(k==0, n, sum(j=1, n, j^k*sumdiv(j, d, 1-(1-1/d)^k)));
%Y A356250 Columns k=0..3 give A001477, A024916, A350123, A356249.
%Y A356250 T(n,n) gives A356238.
%Y A356250 Cf. A344725.
%K A356250 nonn,tabl
%O A356250 1,3
%A A356250 _Seiichi Manyama_, Jul 31 2022