A384454 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th q-factorial number for q=-k.
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 1, -2, -3, 0, 1, 1, 1, -3, -14, 15, 0, 1, 1, 1, -4, -39, 280, 165, 0, 1, 1, 1, -5, -84, 1989, 17080, -3465, 0, 1, 1, 1, -6, -155, 8736, 407745, -3108560, -148995, 0, 1, 1, 1, -7, -258, 28675, 4551456, -333943155, -1700382320, 12664575, 0, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 0, -1, -2, -3, -4, ... 1, 0, -3, -14, -39, -84, ... 1, 0, 15, 280, 1989, 8736, ... 1, 0, 165, 17080, 407745, 4551456, ...
Crossrefs
Programs
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Mathematica
A[n_, k_] := QFactorial[n, -k]; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 10 2025 *) -
PARI
a(n, k) = prod(j=1, n, ((1-(-k)^j)/(1+k)));
Formula
A(n,k) = Product_{j=1..n} (1 - (-k)^j)/(1 + k).