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A015281 Gaussian binomial coefficient [ n,3 ] for q = -12. 2

%I #27 Sep 08 2022 08:44:39

%S 1,-1595,2775445,-4793193515,8283038077141,-14313032243145515,

%T 24732928003956401365,-42738498397393357626155,

%U 73852125402551558141191381,-127616472670861852065241422635

%N Gaussian binomial coefficient [ n,3 ] 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="/A015281/b015281.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 (-1595,231420,2756160,-2985984).

%F a(n) = Product_{i=1..3} ((-12)^(n-i+1)-1)/((-12)^i-1) (by definition). - _Vincenzo Librandi_, Aug 02 2016

%F G.f.: x^3 / ( (x-1)*(12*x+1)*(1728*x+1)*(144*x-1) ). - _R. J. Mathar_, Aug 03 2016

%t Table[QBinomial[n, 3, -12], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)

%o (Sage) [gaussian_binomial(n,3,-12) for n in range(3,13)] # _Zerinvary Lajos_, May 27 2009

%o (Magma) r:=3; q:=-12; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 02 2016

%K sign,easy

%O 3,2

%A _Olivier GĂ©rard_, Dec 11 1999

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Last modified March 29 08:46 EDT 2024. Contains 371267 sequences. (Running on oeis4.)