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%I #21 Sep 08 2022 08:44:39
%S 1,-14762,326882347,-6204226946060,123644349019377043,
%T -2423717068608654822146,47771556642163840723529281,
%U -939857780045414554730512966640,18502040831058043147238631145734166,-364157167636884405223950738210339855212
%N Gaussian binomial coefficient [ n,9 ] 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="/A015375/b015375.txt">Table of n, a(n) for n = 9..200</a>
%F a(n) = Product_{i=1..9} ((-3)^(n-i+1)-1)/((-3)^i-1) (by definition). - _Vincenzo Librandi_, Nov 04 2012
%F G.f.: -x^9 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(19683*x+1)*(6561*x-1)*(243*x+1) ). - _R. J. Mathar_, Sep 02 2016
%t Table[QBinomial[n, 9, -3],{n, 9, 20}] (* _Vincenzo Librandi_, Nov 04 2012 *)
%o (Sage) [gaussian_binomial(n,9,-3) for n in range(9,18)] # _Zerinvary Lajos_, May 25 2009
%o (Magma) r:=9; q:=-3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Nov 04 2012
%Y Cf. Gaussian binomial coefficients [n,9] for q=-2..-13: A015371, A015376, A015377, A015378, A015379, A015380, A015381, A015382, A015383, A015384, A015385. - _Vincenzo Librandi_, Nov 04 2012
%K sign,easy
%O 9,2
%A _Olivier Gérard_
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