A015013 -id:A015013 - cp's OEIS frontend

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

Seiichi Manyama, May 30 2025

			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, ...

Index entries for sequences related to factorial numbers.

Columns k=2..13 give A015013, A015015, A015017, A015018, A015019, A015020, A015022, A015023, A015025, A015026, A015027, A015028.
Main diagonal gives A384453.
Cf. A069777.

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

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

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

A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

Original entry on oeis.org

1, -3, -9, 81, 1215, -40095, -2525985, 325852065, 83092276575, -42626337882975, -43606743654283425, 89350217747626737825, 365889141676531491393375, -2997729737755822508985921375, -49111806293653640164716349886625, 1609344780436736134557590069434814625

Offset: 0

R. J. Mathar, Mar 12 2013

Signed partial products of A062510. This implies that all terms from a(1) on are multiples of 3.

Cf. A005329, A015013, A015109, A027871, A062510, A330862.

A216206 := proc(n)
        mul( (-2)^i-1, i=1..n) ;
end proc:

Table[(-1)^n QPochhammer[-2, -2, n], {n, 0, 15}] (* Bruno Berselli, Mar 13 2013 *)
Table[Product[(-2)^k-1,{k,n}],{n,0,20}] (* Harvey P. Dale, Oct 21 2024 *)

A015109(n,k) = a(n)/(a(k)*a(n-k)).
a(n) = (-3)^n*A015013(n) for n>0, a(0)=1. - Bruno Berselli and Alonso del Arte, Mar 13 2013
a(n) ~ (-1)^(floor(n/2)+1) * c * 2^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/(-2)^k) = 1.21072413030105918013... (A330862). - Amiram Eldar, Aug 10 2025

A269694 Product of first n nonzero Jacobsthal numbers (A001045).

Original entry on oeis.org

1, 1, 3, 15, 165, 3465, 148995, 12664575, 2165642325, 738484032825, 504384594419475, 688484971382583375, 1880252456845835197125, 10268058666835106011499625, 112158004817839862963610403875

Offset: 1

Altug Alkan, Apr 05 2016

Inspired by A015013.

			a(4) = 15 because a(4) = 1*1*3*5 = 15.

Cf. A001045, A015013.

Table[Abs@QFactorial[n, -2], {n, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
FoldList[Times,LinearRecurrence[{1,2},{1,1},20]] (* Harvey P. Dale, Apr 22 2019 *)
Table[(-1)^Floor[n/2] * QPochhammer[-2, 4, 1 + Floor[(n-1)/2]] * QPochhammer[4, 4, Floor[n/2]]/3^n, {n, 1, 20}] (* Vaclav Kotesovec, Mar 04 2021 *)

a001045(n) = (2^n - (-1)^n) / 3;
a(n) = prod(i=1, n, a001045(i));

a(n) = abs(A015013(n)).
a(n) ~ c * 2^(n*(n+1)/2) / 3^n, where c = QPochhammer(-2, 1/4)*QPochhammer(1/4)/3 = 1.21072413030105918013617285610590504636804163112313764347615924554000... - Vaclav Kotesovec, Mar 04 2021, updated Jul 19 2021
Equivalently, c = QPochhammer(-1/2). - Vaclav Kotesovec, Sep 24 2023

cp's OEIS Frontend

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.

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A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

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A269694 Product of first n nonzero Jacobsthal numbers (A001045).

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