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Gaussian binomial coefficient [ n,8 ] for q=-3.
13

%I #24 Sep 08 2022 08:44:39

%S 1,4921,36321901,229798289941,1526550040078063,9974653139743515223,

%T 65533580739687859229563,429769342296322230713871283,

%U 2820146424148466477944423359046,18502040831058043147238631145734166

%N Gaussian binomial coefficient [ n,8 ] 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="/A015357/b015357.txt">Table of n, a(n) for n = 8..200</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (4921,12105660,-8513737740,-2091825362718,169437854380158,4524549298283340,-42209826451809660,-112576695670863081,150094635296999121).

%F a(n) = Product_{i=1..8} ((-3)^(n-i+1)-1)/((-3)^i-1). - _M. F. Hasler_, Nov 03 2012

%F G.f.: -x^8 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(6561*x-1)*(243*x+1) ). - _R. J. Mathar_, Sep 02 2016

%t Table[QBinomial[n, 8, -3], {n, 8, 20}] (* _Vincenzo Librandi_, Nov 02 2012 *)

%o (Sage) [gaussian_binomial(n,8,-3) for n in range(8,18)] # _Zerinvary Lajos_, May 25 2009

%o (Magma) r:=8; q:=-3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // _Vincenzo Librandi_, Nov 02 2012

%o (PARI) A015357(n, r=8, q=-3)=prod(i=1, r, (1-q^(n-i+1))/(1-q^i)) \\ _M. F. Hasler_, Nov 03 2012

%Y Cf. Gaussian binomial coefficients [n,8] for q=-2..-13: A015356, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - _M. F. Hasler_, Nov 03 2012

%K nonn,easy

%O 8,2

%A _Olivier GĂ©rard_