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%I #31 Feb 10 2024 11:34:59
%S 1,-5,55,-385,3311,-25585,208335,-1652145,13275471,-105970865,
%T 848699215,-6785865905,54301841231,-434355079345,3475079247695,
%U -27799679551665,222401254176591,-1779194762447025,14233619183613775
%N Gaussian binomial coefficient [ n,3 ] for q = -2.
%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="/A015266/b015266.txt">Table of n, a(n) for n = 3..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-5, 30, 40, -64).
%F From _Paul Barry_, Jul 12 2005: (Start)
%F G.f.: x^3/((1-2*x-8*x^2)*(1+7*x-8*x^2));
%F a(n) = -5*a(n-1) + 30*a(n-2) + 40*a(n-3) - 64*a(n-4);
%F a(n+3) = (-1)^n*J(n)*J(n+1)*J(n+2)/3, where J(n)=A001045(n). (End)
%F a(n) = T015109(n,3), where T015109 is the triangular array defined by A015109. - _M. F. Hasler_, Nov 04 2012
%t Table[QBinomial[n, 2, -2], {n, 3, 25}] (* _G. C. Greubel_, Jul 31 2016 *)
%o (Sage) [gaussian_binomial(n,3,-2) for n in range(3,22)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) [(1/81)*(24*4^n-6*(-2)^n+64*(-8)^n-1): n in [0..20]]; // _Vincenzo Librandi_, Aug 23 2011
%Y Diagonal k=3 of the triangular array A015109. See there for further references and programs. - _M. F. Hasler_, Nov 04 2012
%K sign,easy
%O 3,2
%A _Olivier Gérard_, Dec 11 1999
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