A292166 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j).

1, 1, -1, 1, -1, -1, 1, -1, -2, 0, 1, -1, -4, -1, 0, 1, -1, -8, -5, -1, 1, 1, -1, -16, -19, -7, 5, 0, 1, -1, -32, -65, -37, 27, 1, 1, 1, -1, -64, -211, -175, 155, 17, 13, 0, 1, -1, -128, -665, -781, 927, 205, 167, 4, 0, 1, -1, -256, -2059, -3367, 5675, 2129, 2089, 110, 0, 0

Offset: 0

Author

Seiichi Manyama, Sep 10 2017

Keywords

• sign
• look
• tabl

Examples

			Square array begins:
    1,  1,  1,   1,    1, ...
   -1, -1, -1,  -1,   -1, ...
   -1, -2, -4,  -8,  -16, ...
    0, -1, -5, -19,  -65, ...
    0, -1, -7, -37, -175, ...
		

Links

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

Crossrefs

Columns k=0..2 give A010815, A022661, A292164.

Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000079.

Main diagonal gives A292167.

Programs

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, -(1/n)*Sum[Sum[d^(1+k*j/d), {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 04 2022 *)

Formula

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017