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)).

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, 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, -5, 0, -3, 4, 0, 0, 1, -1, 1, -3, 4, -5, 0, -7, -2, -2, 0, 0, 1, -1, 1

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, -3, -3, ...
   0,  0,  0,  0,  4, ...
   0,  1, -3,  0,  0, ...

Mathematica

nmax = 12;
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]&;
A[n_, k_] := If[n == 0, 1, If[k == 0, 0, col[k][[n+1]]]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 21 2021 *)

Crossrefs

Columns k=0..2 give A000007, A010815, A293072.

Rows n=0 gives A000012.

Main diagonal gives A293306.

Cf. A293071, A293307.

Sequence in context: A323878 A046094 A055976 * A316896 A230626 A363946

Adjacent sequences: A293302 A293303 A293304 * A293306 A293307 A293308

Keyword

sign,tabl,look

Author

Seiichi Manyama, Oct 05 2017

Status

approved