%I #25 Sep 08 2022 08:44:39
%S 1,57,3705,236665,15150201,969583737,62053592185,3971428035705,
%T 254171409198201,16266970069380217,1041086085394771065,
%U 66629509457629850745,4264288605349394427001,272914470741872571493497
%N Gaussian binomial coefficient [ n,2 ] 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="/A015259/b015259.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 (57,456,-512).
%F G.f.: x^2/((1-x)*(1+8*x)*(1-64*x)).
%F a(2) = 1, a(3) = 57, a(4) = 3705, a(n) = 57*a(n-1) + 456*a(n-2) - 512*a(n-3). - _Vincenzo Librandi_, Oct 27 2012
%t Table[QBinomial[n, 2, -8], {n, 2, 20}] (* _Vincenzo Librandi_, Oct 27 2012 *)
%o (Sage) [gaussian_binomial(n,2,-8) for n in range(2,16)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) I:=[1, 57, 3705]; [n le 3 select I[n] else 57*Self(n-1)+456*Self(n-2)-512*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Oct 27 2012
%K nonn,easy
%O 2,2
%A _Olivier GĂ©rard_, Dec 11 1999
|