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A321876 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - x^j)^sigma_k(j).

%I A321876 #13 Nov 23 2018 04:07:14
%S A321876 1,1,1,1,1,3,1,1,4,5,1,1,6,8,11,1,1,10,16,21,17,1,1,18,38,52,39,34,1,
%T A321876 1,34,100,156,128,92,52,1,1,66,278,526,534,373,170,94,1,1,130,796,
%U A321876 1896,2546,2014,913,360,145,1,1,258,2318,7102,13074,12953,6796,2399,667,244
%N A321876 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - x^j)^sigma_k(j).
%H A321876 Seiichi Manyama, <a href="/A321876/b321876.txt">Antidiagonals n = 0..139, flattened</a>
%F A321876 G.f. of column k: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(j^k).
%F A321876 G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^j))).
%e A321876 Square array begins:
%e A321876    1,   1,    1,    1,     1,      1,  ...
%e A321876    1,   1,    1,    1,     1,      1,  ...
%e A321876    3,   4,    6,   10,    18,     34,  ...
%e A321876    5,   8,   16,   38,   100,    278,  ...
%e A321876   11,  21,   52,  156,   526,   1896,  ...
%e A321876   17,  39,  128,  534,  2546,  13074,  ...
%t A321876 Table[Function[k, SeriesCoefficient[Product[1/(1 - x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
%t A321876 Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^j)), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
%Y A321876 Columns k=0..9 give A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547.
%Y A321876 Main diagonal gives A319647.
%Y A321876 Cf. A321877.
%K A321876 nonn,tabl
%O A321876 0,6
%A A321876 _Ilya Gutkovskiy_, Nov 20 2018