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A293305 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} (-1)^j*j*x^(j*i)).

%I A293305 #23 Feb 21 2021 06:40:20
%S A293305 1,1,0,1,-1,0,1,-1,-1,0,1,-1,1,0,0,1,-1,1,0,0,0,1,-1,1,-3,0,1,0,1,-1,
%T A293305 1,-3,0,-3,0,0,1,-1,1,-3,4,0,4,1,0,1,-1,1,-3,4,0,4,-3,0,0,1,-1,1,-3,4,
%U A293305 -5,0,-3,4,0,0,1,-1,1,-3,4,-5,0,-7,-2,-2,0,0,1,-1,1
%N A293305 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} (-1)^j*j*x^(j*i)).
%H A293305 Seiichi Manyama, <a href="/A293305/b293305.txt">Antidiagonals n = 0..139, flattened</a>
%e A293305 Square array begins:
%e A293305    1,  1,  1,  1,  1, ...
%e A293305    0, -1, -1, -1, -1, ...
%e A293305    0, -1,  1,  1,  1, ...
%e A293305    0,  0,  0, -3, -3, ...
%e A293305    0,  0,  0,  0,  4, ...
%e A293305    0,  1, -3,  0,  0, ...
%t A293305 nmax = 12;
%t A293305 col[k_] := col[k] = Product[1+Sum[(-1)^j*j*x^(i*j), {j, 1, k}], {i, 1, 2 nmax}] + O[x]^(2 nmax) // CoefficientList[#, x]&;
%t A293305 A[n_, k_] := If[n == 0, 1, If[k == 0, 0, col[k][[n+1]]]];
%t A293305 Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Feb 21 2021 *)
%Y A293305 Columns k=0..2 give A000007, A010815, A293072.
%Y A293305 Rows n=0 gives A000012.
%Y A293305 Main diagonal gives A293306.
%Y A293305 Cf. A293071, A293307.
%K A293305 sign,tabl,look
%O A293305 0,25
%A A293305 _Seiichi Manyama_, Oct 05 2017