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A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

%I A302233 #5 Mar 29 2019 15:51:20
%S A302233 1,1,0,1,-1,0,1,-1,1,0,1,-1,0,-2,0,1,-1,0,0,2,0,1,-1,0,-1,0,-3,0,1,-1,
%T A302233 0,-1,2,-1,4,0,1,-1,0,-1,1,-2,1,-5,0,1,-1,0,-1,1,0,1,-1,6,0,1,-1,0,-1,
%U A302233 1,-1,0,-2,1,-8,0,1,-1,0,-1,1,-1,2,-1,4,0,10,0,1,-1,0,-1,1,-1,1,-2,1,-4,0,-12,0
%N A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).
%F A302233 G.f. of column k: Product_{j>=1} (1 + x^(k*j))/(1 + x^j).
%F A302233 For asymptotics of column k see comment from _Vaclav Kotesovec_ in A145707.
%e A302233 Square array begins:
%e A302233 1,  1,  1,  1,  1,  1,  ...
%e A302233 0, -1, -1, -1, -1, -1,  ...
%e A302233 0,  1,  0,  0,  0,  0,  ...
%e A302233 0, -2,  0, -1, -1, -1,  ...
%e A302233 0,  2,  0,  2,  1,  1,  ...
%e A302233 0, -3, -1, -2,  0, -1,  ...
%t A302233 Table[Function[k, SeriesCoefficient[Product[(1 + x^(k i))/(1 + x^i), {i, 1, n}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
%t A302233 Table[Function[k, SeriesCoefficient[QPochhammer[-1, x^k]/QPochhammer[-1, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
%Y A302233 Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.
%Y A302233 Main diagonal gives A081362.
%Y A302233 Cf. A286653, A286656, A290307.
%K A302233 sign,tabl
%O A302233 0,14
%A A302233 _Ilya Gutkovskiy_, Apr 03 2018