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A356045 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} sigma_k(j) * floor(n/j).

%I A356045 #21 Jul 25 2022 15:41:31
%S A356045 1,1,4,1,5,7,1,7,10,13,1,11,18,21,16,1,19,40,45,28,25,1,35,102,123,72,
%T A356045 48,28,1,67,280,393,250,138,57,38,1,131,798,1371,1020,540,189,83,44,1,
%U A356045 259,2320,5025,4498,2514,885,301,101,53,1,515,6822,18963,20652,12828,4917,1553,403,129,56
%N A356045 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} sigma_k(j) * floor(n/j).
%H A356045 Seiichi Manyama, <a href="/A356045/b356045.txt">Antidiagonals n = 1..140, flattened</a>
%F A356045 G.f. of column k: (1/(1-x)) * Sum_{j>=1} sigma_k(j) * x^j/(1 - x^j).
%F A356045 T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k * tau(j/d).
%F A356045 T(n,k) = Sum_{j=1..n} Sum_{d|j} sigma_k(d).
%e A356045 Square array begins:
%e A356045    1,  1,   1,   1,    1,     1, ...
%e A356045    4,  5,   7,  11,   19,    35, ...
%e A356045    7, 10,  18,  40,  102,   280, ...
%e A356045   13, 21,  45, 123,  393,  1371, ...
%e A356045   16, 28,  72, 250, 1020,  4498, ...
%e A356045   25, 48, 138, 540, 2514, 12828, ...
%o A356045 (PARI) T(n, k) = sum(j=1, n, sigma(j, k)*(n\j));
%o A356045 (PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, d^k*numdiv(j/d)));
%o A356045 (PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, sigma(d, k)));
%Y A356045 Columns k=0..3 give A061201, A280077, A356042, A356043.
%Y A356045 T(n,n) gives A356046.
%Y A356045 Cf. A322103.
%K A356045 nonn,tabl
%O A356045 1,3
%A A356045 _Seiichi Manyama_, Jul 24 2022