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

0,18

Links

Table of n, a(n) for n=0..65.

Index entries for sequences related to factorial numbers.

Formula

A(n,k) = Product_{j=1..n} (1 - (-k)^j)/(1 + k).

Example

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, ...

Mathematica

A[n_, k_] := QFactorial[n, -k]; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 10 2025 *)

Prog

(PARI) a(n, k) = prod(j=1, n, ((1-(-k)^j)/(1+k)));

Crossrefs

Columns k=2..13 give A015013, A015015, A015017, A015018, A015019, A015020, A015022, A015023, A015025, A015026, A015027, A015028.

Main diagonal gives A384453.

Cf. A069777.

Sequence in context: A375580 A030386 A096799 * A370292 A243081 A287847

Adjacent sequences: A384451 A384452 A384453 * A384455 A384456 A384457

Keyword

sign,tabl

Author

Seiichi Manyama, May 30 2025

Status

approved