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).

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

Offset

0,25

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Example

Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0, -1,  1,  1,  1, ...
   0,  0,  0, -6, -6, ...
   0,  0,  0,  0, 24, ...
   0,  1, -3,  3,  3, ...

Mathematica

nmax = 12;
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;
A[n_, k_] := M[[n+1, k+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2020 *)

Crossrefs

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

Rows n=0 gives A000012.

Main diagonal gives A293236.

Cf. A293202.

Sequence in context: A256041 A137378 A333275 * A084680 A051626 A357003

Adjacent sequences: A293068 A293069 A293070 * A293072 A293073 A293074

Keyword

sign,tabl,look

Author

Seiichi Manyama, Oct 03 2017

Status

approved