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A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

%I A322103 #12 Jul 26 2022 10:09:38
%S A322103 1,1,3,1,4,3,1,6,5,6,1,10,11,11,3,1,18,29,27,7,9,1,34,83,83,27,20,3,1,
%T A322103 66,245,291,127,66,9,10,1,130,731,1091,627,290,51,26,6,1,258,2189,
%U A322103 4227,3127,1494,345,112,18,9,1,514,6563,16643,15627,8330,2403,668,102,28,3
%N A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).
%H A322103 Seiichi Manyama, <a href="/A322103/b322103.txt">Antidiagonals n = 1..140, flattened</a>
%F A322103 G.f. of column k: Sum_{j>=1} sigma_k(j)*x^j/(1 - x^j).
%F A322103 A(n,k) = Sum_{d|n} d^k*tau(n/d).
%e A322103 Square array begins:
%e A322103   1,   1,   1,    1,     1,     1,  ...
%e A322103   3,   4,   6,   10,    18,    34,  ...
%e A322103   3,   5,  11,   29,    83,   245,  ...
%e A322103   6,  11,  27,   83,   291,  1091,  ...
%e A322103   3,   7,  27,  127,   627,  3127,  ...
%e A322103   9,  20,  66,  290,  1494,  8330,  ...
%t A322103 Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%t A322103 Table[Function[k, SeriesCoefficient[Sum[DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
%o A322103 (PARI) T(n,k)={sumdiv(n, d, d^k*numdiv(n/d))}
%o A322103 for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 26 2018
%Y A322103 Columns k=0..3 give A007425, A007429, A007433, A321140.
%Y A322103 Cf. A109974, A321141 (diagonal), A356045.
%K A322103 nonn,tabl
%O A322103 1,3
%A A322103 _Ilya Gutkovskiy_, Nov 26 2018