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A292166 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*x^j).

%I A292166 #30 Sep 04 2022 17:10:23
%S A292166 1,1,-1,1,-1,-1,1,-1,-2,0,1,-1,-4,-1,0,1,-1,-8,-5,-1,1,1,-1,-16,-19,
%T A292166 -7,5,0,1,-1,-32,-65,-37,27,1,1,1,-1,-64,-211,-175,155,17,13,0,1,-1,
%U A292166 -128,-665,-781,927,205,167,4,0,1,-1,-256,-2059,-3367,5675,2129,2089,110,0,0
%N A292166 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*x^j).
%H A292166 Seiichi Manyama, <a href="/A292166/b292166.txt">Antidiagonals n = 0..139, flattened</a>
%F A292166 A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - _Seiichi Manyama_, Nov 02 2017
%e A292166 Square array begins:
%e A292166     1,  1,  1,   1,    1, ...
%e A292166    -1, -1, -1,  -1,   -1, ...
%e A292166    -1, -2, -4,  -8,  -16, ...
%e A292166     0, -1, -5, -19,  -65, ...
%e A292166     0, -1, -7, -37, -175, ...
%t A292166 A[n_, k_] := A[n, k] = If[n == 0, 1, -(1/n)*Sum[Sum[d^(1+k*j/d), {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]];
%t A292166 Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Sep 04 2022 *)
%Y A292166 Columns k=0..2 give A010815, A022661, A292164.
%Y A292166 Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000079.
%Y A292166 Main diagonal gives A292167.
%K A292166 sign,look,tabl
%O A292166 0,9
%A A292166 _Seiichi Manyama_, Sep 10 2017