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A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

%I A350106 #20 Dec 15 2021 07:52:54
%S A350106 1,1,4,1,6,8,1,10,14,15,1,18,32,31,21,1,34,86,87,45,33,1,66,248,295,
%T A350106 153,81,41,1,130,734,1095,669,309,101,56,1,258,2192,4231,3201,1521,
%U A350106 443,150,69,1,514,6566,16647,15765,8373,2633,722,191,87,1,1026,19688,66055,78393,48321,17411,4746,1005,253,99
%N A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.
%H A350106 Seiichi Manyama, <a href="/A350106/b350106.txt">Antidiagonals n = 1..140, flattened</a>
%F A350106 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j)^2.
%F A350106 T(n,k) = Sum_{j=1..n} j * Sum_{d|j} (d^k - (d - 1)^k)/d.
%e A350106 Square array begins:
%e A350106    1,   1,   1,    1,     1,      1,      1, ...
%e A350106    4,   6,  10,   18,    34,     66,    130, ...
%e A350106    8,  14,  32,   86,   248,    734,   2192, ...
%e A350106   15,  31,  87,  295,  1095,   4231,  16647, ...
%e A350106   21,  45, 153,  669,  3201,  15765,  78393, ...
%e A350106   33,  81, 309, 1521,  8373,  48321, 284709, ...
%e A350106   41, 101, 443, 2633, 17411, 119321, 828323, ...
%t A350106 T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Dec 14 2021 *)
%o A350106 (PARI) T(n, k) = sum(j=1, n, j*(n\j)^k);
%o A350106 (PARI) T(n, k) = sum(j=1, n, j*sumdiv(j, d, (d^k-(d-1)^k)/d));
%Y A350106 Columns k=1..3 give A024916, A350107, A350108.
%Y A350106 T(n,n) gives A350109.
%Y A350106 Cf. A319649, A344725.
%K A350106 nonn,tabl
%O A350106 1,3
%A A350106 _Seiichi Manyama_, Dec 14 2021