A293071 - OEIS

%I #23 Nov 15 2020 09:00:25

%S 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 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 24,-117,-20,-3,4,0,0,1,-1,1,-6,24,-117,-20,-27,-8,-2

%N 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 Seiichi Manyama, <a href="/A293071/b293071.txt">Antidiagonals n = 0..139, flattened</a>

%e Square array begins:

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

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

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

%e    0,  0,  0, -6, -6, ...

%e    0,  0,  0,  0, 24, ...

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

%t nmax = 12;

%t 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 A[n_, k_] := M[[n+1, k+1]];

%t Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 15 2020 *)

%Y Columns k=0..5 give A000007, A010815, A293072, A293255, A293256, A293257.

%Y Rows n=0 gives A000012.

%Y Main diagonal gives A293236.

%Y Cf. A293202.

%K sign,tabl,look

%O 0,25

%A _Seiichi Manyama_, Oct 03 2017