|
%I #24 Sep 08 2022 08:44:39
%S 1,-20,610,-15860,433771,-11662040,315323620,-8509702520,229798289941,
%T -6204226946060,167517069529030,-4522934399547980,122119467087816511,
%U -3297223466672052080,89025052902439936840,-2403676254645238280240
%N Gaussian binomial coefficient [ n,3 ] for q = -3.
%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="/A015268/b015268.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 (-20,210,540,-729).
%F G.f.: x^3/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)). - _Bruno Berselli_, Oct 29 2012
%F a(n) = (-1 + 7*3^(2n-3) + (-1)^n*3^(n-2)*(7-3^(2n-1)))/896. - _Bruno Berselli_, Oct 29 2012
%t Table[QBinomial[n, 3, -3], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)
%o (Sage) [gaussian_binomial(n,3,-3) for n in range(3,19)] # _Zerinvary Lajos_, May 27 2009
%o (Magma) [(-1+7*3^(2*n-3)+(-1)^n*3^(n-2)*(7-3^(2*n-1)))/896: n in [3..18]]; // _Bruno Berselli_, Oct 29 2012
%o (Maxima) makelist(coeff(taylor(1/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)), x, 0, n), x, n), n, 0, 15); \\ _Bruno Berselli_, Oct 29 2012
%K sign,easy
%O 3,2
%A _Olivier Gérard_, Dec 11 1999
|
