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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.

%I A384454 #25 Aug 10 2025 02:36:54
%S A384454 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,
%T A384454 -4,-39,280,165,0,1,1,1,-5,-84,1989,17080,-3465,0,1,1,1,-6,-155,8736,
%U A384454 407745,-3108560,-148995,0,1,1,1,-7,-258,28675,4551456,-333943155,-1700382320,12664575,0,1
%N 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.
%H A384454 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A384454 A(n,k) = Product_{j=1..n} (1 - (-k)^j)/(1 + k).
%e A384454 Square array begins:
%e A384454   1, 1,   1,     1,      1,       1, ...
%e A384454   1, 1,   1,     1,      1,       1, ...
%e A384454   1, 0,  -1,    -2,     -3,      -4, ...
%e A384454   1, 0,  -3,   -14,    -39,     -84, ...
%e A384454   1, 0,  15,   280,   1989,    8736, ...
%e A384454   1, 0, 165, 17080, 407745, 4551456, ...
%t A384454 A[n_, k_] := QFactorial[n, -k]; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Aug 10 2025 *)
%o A384454 (PARI) a(n, k) = prod(j=1, n, ((1-(-k)^j)/(1+k)));
%Y A384454 Columns k=2..13 give A015013, A015015, A015017, A015018, A015019, A015020, A015022, A015023, A015025, A015026, A015027, A015028.
%Y A384454 Main diagonal gives A384453.
%Y A384454 Cf. A069777.
%K A384454 sign,tabl
%O A384454 0,18
%A A384454 _Seiichi Manyama_, May 30 2025