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A294653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j) in powers of x.

%I A294653 #21 Nov 10 2017 05:55:10
%S A294653 1,1,-1,1,-1,-1,1,-1,-4,0,1,-1,-16,-23,0,1,-1,-64,-713,-229,1,1,-1,
%T A294653 -256,-19619,-64807,-2761,0,1,-1,-1024,-531185,-16757533,-9688425,
%U A294653 -42615,1,1,-1,-4096,-14347883,-4294435855,-30499541197,-2165979799,-758499,0
%N A294653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j) in powers of x.
%H A294653 Seiichi Manyama, <a href="/A294653/b294653.txt">Antidiagonals n = 0..52, flattened</a>
%F A294653 A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j)) * A(n-j,k) for n > 0.
%e A294653 Square array begins:
%e A294653     1,    1,      1,         1,           1, ...
%e A294653    -1,   -1,     -1,        -1,          -1, ...
%e A294653    -1,   -4,    -16,       -64,        -256, ...
%e A294653     0,  -23,   -713,    -19619,     -531185, ...
%e A294653     0, -229, -64807, -16757533, -4294435855, ...
%t A294653 rows = 10;
%t A294653 col[k_] := col[k] = CoefficientList[Product[(1 - j^(k*j)*x^j), {j, 1, rows + 3}] + O[x]^(rows + 3), x];
%t A294653 A[n_, k_] := col[k][[n + 1]];
%t A294653 (* or: *)
%t A294653 A[0, _] = 1; A[n_, k_] := A[n, k] = -(1/n)*Sum[DivisorSum[j, #^(1 + k*j) &]*A[n - j, k], {j, 1, n}];
%t A294653 Table[A[n - k, k], {n, 0, rows - 1}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 10 2017 *)
%Y A294653 Columns k=0..1 give A010815, A292312.
%Y A294653 Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000302.
%Y A294653 Cf. A283675, A294758.
%K A294653 sign,tabl
%O A294653 0,9
%A A294653 _Seiichi Manyama_, Nov 06 2017