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A293071 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} (-1)^j*j!*x^(j*i).

%I A293071 #23 Nov 15 2020 09:00:25
%S A293071 1,1,0,1,-1,0,1,-1,-1,0,1,-1,1,0,0,1,-1,1,0,0,0,1,-1,1,-6,0,1,0,1,-1,
%T A293071 1,-6,0,-3,0,0,1,-1,1,-6,24,3,4,1,0,1,-1,1,-6,24,3,4,-3,0,0,1,-1,1,-6,
%U A293071 24,-117,-20,-3,4,0,0,1,-1,1,-6,24,-117,-20,-27,-8,-2
%N A293071 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} (-1)^j*j!*x^(j*i).
%H A293071 Seiichi Manyama, <a href="/A293071/b293071.txt">Antidiagonals n = 0..139, flattened</a>
%e A293071 Square array begins:
%e A293071    1,  1,  1,  1,  1, ...
%e A293071    0, -1, -1, -1, -1, ...
%e A293071    0, -1,  1,  1,  1, ...
%e A293071    0,  0,  0, -6, -6, ...
%e A293071    0,  0,  0,  0, 24, ...
%e A293071    0,  1, -3,  3,  3, ...
%t A293071 nmax = 12;
%t A293071 col[k_] := CoefficientList[Product[Sum[(-1)^j j! x^(i j), {j, 0, k}], {i, 1, nmax+1}] + O[x]^(nmax+1), x]; M = PadRight[col[#], nmax+1]& /@ Range[0, nmax] // Transpose;
%t A293071 A[n_, k_] := M[[n+1, k+1]];
%t A293071 Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 15 2020 *)
%Y A293071 Columns k=0..5 give A000007, A010815, A293072, A293255, A293256, A293257.
%Y A293071 Rows n=0 gives A000012.
%Y A293071 Main diagonal gives A293236.
%Y A293071 Cf. A293202.
%K A293071 sign,tabl,look
%O A293071 0,25
%A A293071 _Seiichi Manyama_, Oct 03 2017