A015013 q-factorial numbers for q=-2.

1, 1, -1, -3, 15, 165, -3465, -148995, 12664575, 2165642325, -738484032825, -504384594419475, 688484971382583375, 1880252456845835197125, -10268058666835106011499625, -112158004817839862963610403875, 2450091615245711806440069272649375, 107046952761700394535173066591323843125

Offset: 0

Author

Olivier Gérard

Keywords

• sign
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..80
• Index entries for sequences related to factorial numbers.

Crossrefs

Column k=2 of A384454.

Cf. A001045, A330862.

Programs

Magma

I:=[1]; [n le 1 select I[n] else (((-2)^n - 1) * Self(n-1))/(-3): n in [1..18]]; // Vincenzo Librandi, Oct 26 2012

Mathematica

RecurrenceTable[{a[1]==1, a[n]==(((-2)^n - 1) * a[n-1])/(-3)}, a, {n, 15}]
Table[QFactorial[n, -2], {n, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

PARI

a(n) = prod(k=1, n, ((-2)^k-1)/(-3)) \\ Michel Marcus, Apr 05 2016

Formula

a(n) = Product_{k=1..n} ((-2)^k - 1) / (-2 - 1).

a(1) = 1, a(n) = (((-2)^n - 1) * a(n-1))/(-3). - Vincenzo Librandi, Oct 26 2012

a(n) = (-1)^(floor((n mod 4)/2)) * Product_{k=1..n} A001045(k). - Altug Alkan, Apr 05 2016

a(n) ~ (-1)^floor(n/2) * c * 2^(n*(n+1)/2) / 3^n, where c = Product_{k>=1} (1 - 1/(-2)^k) = 1.21072413030105918013... (A330862). - Amiram Eldar, Aug 09 2025

Extensions

a(0)=1 prepended by Seiichi Manyama, May 30 2025