A302233 - OEIS

%I #5 Mar 29 2019 15:51:20

%S 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 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 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 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 G.f. of column k: Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

%F For asymptotics of column k see comment from _Vaclav Kotesovec_ in A145707.

%e Square array begins:

%e 1,  1,  1,  1,  1,  1,  ...

%e 0, -1, -1, -1, -1, -1,  ...

%e 0,  1,  0,  0,  0,  0,  ...

%e 0, -2,  0, -1, -1, -1,  ...

%e 0,  2,  0,  2,  1,  1,  ...

%e 0, -3, -1, -2,  0, -1,  ...

%t 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 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 Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.

%Y Main diagonal gives A081362.

%Y Cf. A286653, A286656, A290307.

%K sign,tabl

%O 0,14

%A _Ilya Gutkovskiy_, Apr 03 2018