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q-factorial numbers for q=-2.
3

%I #29 Sep 08 2022 08:44:39

%S 1,-1,-3,15,165,-3465,-148995,12664575,2165642325,-738484032825,

%T -504384594419475,688484971382583375,1880252456845835197125,

%U -10268058666835106011499625,-112158004817839862963610403875

%N q-factorial numbers for q=-2.

%H Vincenzo Librandi, <a href="/A015013/b015013.txt">Table of n, a(n) for n = 1..80</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

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

%F a(1) = 1, a(n) = (((-2)^n - 1) * a(n-1))/(-3). - _Vincenzo Librandi_, Oct 26 2012

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

%t RecurrenceTable[{a[1]==1, a[n]==(((-2)^n - 1) * a[n-1])/(-3)}, a, {n, 15}]

%t Table[QFactorial[n, -2], {n, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)

%o (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

%o (PARI) a(n) = prod(k=1, n, ((-2)^k-1)/(-3)) \\ _Michel Marcus_, Apr 05 2016

%Y Cf. A001045.

%K sign,easy

%O 1,3

%A _Olivier GĂ©rard_