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A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

%I A290307 #11 Mar 29 2019 15:51:20
%S A290307 1,1,0,1,1,0,1,1,0,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,2,1,1,0,1,1,1,2,1,2,
%T A290307 1,0,1,1,1,2,2,2,2,1,0,1,1,1,2,2,2,3,3,2,0,1,1,1,2,2,3,3,3,3,2,0,1,1,
%U A290307 1,2,2,3,3,4,4,3,2,0,1,1,1,2,2,3,4,4,4,5,4,2,0,1,1,1,2,2,3,4,4,5,6,6,5,3,0
%N A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).
%C A290307 A(n,k) is the number of partitions of n into distinct parts where no part is a multiple of k.
%H A290307 Seiichi Manyama, <a href="/A290307/b290307.txt">Antidiagonals n = 0..139, flattened</a>
%H A290307 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A290307 G.f. of column k: Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).
%F A290307 For asymptotics of column k see comment from _Vaclav Kotesovec_ in A261772.
%e A290307 Square array begins:
%e A290307   1,  1,  1,  1,  1,  1, ...
%e A290307   0,  1,  1,  1,  1,  1, ...
%e A290307   0,  0,  1,  1,  1,  1, ...
%e A290307   0,  1,  1,  2,  2,  2, ...
%e A290307   0,  1,  1,  1,  2,  2, ...
%e A290307   0,  1,  2,  2,  2,  3, ...
%t A290307 Table[Function[k, SeriesCoefficient[Product[(1 + x^i)/(1 + x^(i k)), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
%t A290307 Table[Function[k, SeriesCoefficient[QPochhammer[-1, x]/QPochhammer[-1, x^k], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
%Y A290307 Columns k=1-10 give: A000007, A000700, A003105, A070048, A096938, A261770, A097793, A261771, A112193, A261772.
%Y A290307 Cf. A286653.
%K A290307 nonn,tabl
%O A290307 0,25
%A A290307 _Ilya Gutkovskiy_, Jul 26 2017