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A015273 Gaussian binomial coefficient [ n,3 ] for q=-6.

%I A015273 #40 Sep 08 2022 08:44:39
%S A015273 1,-185,41107,-8838005,1910490043,-412612541285,89126228045659,
%T A015273 -19251196169490725,4158260859792814555,-898184256176675135525,
%U A015273 194007802557550502202331,-41905685236388916561230885
%N A015273 Gaussian binomial coefficient [ n,3 ] for q=-6.
%C A015273 From _Bruno Berselli_, Oct 30 2012: (Start)
%C A015273 More generally, for sequences of the type "Gaussian binomial coefficient [n,3] for q=-m", we have:
%C A015273 a(n) = (1-(-m)^n)*(1-(-m)^(n-1))*(1-(-m)^(n-2))/((1+m)*(1-m^2)*(1+m^3)) = (-1+(1-m+m^2)*m^(2n-3)+(-1)^n*m^(n-2)*(1-m+m^2-m^(2n-1)))/(-1-m+m^2-m^4+m^5+m^6),
%C A015273 G.f.: x^3/((1-x)*(1+m*x)*(1-m^2*x)*(1+m^3*x)). (End)
%D A015273 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 A015273 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A015273 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A015273 Vincenzo Librandi, <a href="/A015273/b015273.txt">Table of n, a(n) for n = 3..200</a>
%H A015273 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-185,6882,39960,-46656).
%F A015273 G.f.: x^3/((1-x)*(1+6*x)*(1-36*x)*(1+216*x)). - _Bruno Berselli_, Oct 30 2012
%F A015273 a(n) = (-1+31*6^(2n-3)+(-1)^n*6^(n-2)*(31-6^(2n-1)))/53165. - _Bruno Berselli_, Oct 30 2012
%t A015273 Table[QBinomial[n, 3, -6], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)
%o A015273 (Sage) [gaussian_binomial(n,3,-6) for n in range(3,15)] # _Zerinvary Lajos_, May 27 2009
%o A015273 (Magma) I:=[1,-185,41107,-8838005]; [n le 4 select I[n] else -185*Self(n-1)+6882*Self(n-2)+39960*Self(n-3)-46656*Self(n-4): n in [1..13]]; // _Vincenzo Librandi_, Oct 29 2012
%K A015273 sign,easy
%O A015273 3,2
%A A015273 _Olivier Gérard_, Dec 11 1999