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%I #29 Sep 08 2022 08:44:39
%S 1,-300,105050,-35927100,12328144851,-4228301370600,1450319733570100,
%T -497459062806004200,170628488227082949701,-58525570007342935110900,
%U 20074270583791406305395150,-6885474806748086165925231300
%N Gaussian binomial coefficient [ n,3 ] for q = -7.
%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="/A015275/b015275.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 (-300,15050,102900,-117649).
%F G.f.: x^3/((1-x)*(1+7*x)*(1-49*x)*(1+343*x)). - _Bruno Berselli_, Oct 30 2012
%F a(n) = (-1 + 43*7^(2n-3) + (-1)^n*7^(n-2)*(43-7^(2n-1)))/132096. - _Bruno Berselli_, Oct 30 2012
%t QBinomial[Range[3,20],3,-7] (* _Harvey P. Dale_, Apr 09 2012 *)
%t Table[QBinomial[n, 3, -7], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)
%o (Sage) [gaussian_binomial(n,3,-7) for n in range(3,15)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) r:=3; q:=-7; [&*[(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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