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%I #28 Sep 08 2022 08:44:39
%S 1,-455,236665,-120935815,61934287481,-31709385606535,
%T 16235267484138105,-8312452980450674055,4255976180162154314361,
%U -2179059787976052939572615,1115678612484825190455949945,-571227449525600988055816521095
%N Gaussian binomial coefficient [ n,3 ] for q = -8.
%D J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H Vincenzo Librandi, <a href="/A015276/b015276.txt">Table of n, a(n) for n = 3..200</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-455,29640,232960,-262144).
%F G.f.: x^3/((1-x)*(1+8*x)*(1-64*x)*(1+512*x)). - _Bruno Berselli_, Oct 30 2012
%F a(n) = (-1 + 57*8^(2n-3) + (-1)^n*8^(n-2)*(57-8^(2n-1)))/290871. - _Bruno Berselli_, Oct 30 2012
%F a(n) = Product_{i=1..3} ((-8)^(n-i+1)-1)/((-8)^i-1) (by definition). - _Vincenzo Librandi_, Aug 02 2016
%t Table[QBinomial[n, 3, -8], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)
%o (Sage) [gaussian_binomial(n,3,-8) for n in range(3,15)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) r:=3; q:=-8; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 02 2016
%K sign,easy
%O 3,2
%A _Olivier Gérard_, Dec 11 1999
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