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Gaussian binomial coefficient [ n,2 ] for q = -12.
3

%I #28 Sep 08 2022 08:44:39

%S 1,133,19285,2775445,399683221,57554154133,8287800951445,

%T 1193443303932565,171855836163195541,24747240402737283733,

%U 3563602618051323347605,513158776998704708174485,73894863887821708223693461

%N Gaussian binomial coefficient [ n,2 ] for q = -12.

%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="/A015264/b015264.txt">Table of n, a(n) for n = 2..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (133, 1596, -1728).

%F G.f.: x^2/((1-x)*(1+12*x)*(1-144*x)).

%F a(2) = 1, a(3) = 133, a(4) = 19285, a(n) = 133*a(n-1) + 1596*a(n-2) - 1728*a(n-3). - _Vincenzo Librandi_, Oct 28 2012

%t Table[QBinomial[n, 2, -12], {n, 2,20}] (* _Vincenzo Librandi_, Oct 28 2012 *)

%o (Sage) [gaussian_binomial(n,2,-12) for n in range(2,14)] # _Zerinvary Lajos_, May 27 2009

%o (Magma) I:=[1,133,19285]; [n le 3 select I[n] else 133*Self(n-1)+1596*Self(n-2)-1728*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Oct 28 2012

%K nonn,easy

%O 2,2

%A _Olivier GĂ©rard_, Dec 11 1999