login
Gaussian binomial coefficient [ n,3 ] for q = -10.
2

%I #28 Sep 08 2022 08:44:39

%S 1,-909,918191,-917272809,917364637191,-917355454462809,

%T 917356372736537191,-917356280909173462809,917356290091909926537191,

%U -917356289173636281073462809,917356289265463645628926537191

%N Gaussian binomial coefficient [ n,3 ] for q = -10.

%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="/A015278/b015278.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 (-909,91910,909000,-1000000).

%F G.f.: x^3/((1-x)*(1+10*x)*(1-100*x)*(1+1000*x)). - _Bruno Berselli_, Oct 30 2012

%F a(n) = (-1 + 91*10^(2n-3) + (-1)^n*10^(n-2)*(91-10^(2n-1)))/1090089. - _Bruno Berselli_, Oct 30 2012

%F a(n) = product(((-10)^(n-i+1)-1)/((-10)^i-1), i=1..3) (by definition). - _Vincenzo Librandi_, Aug 02 2016

%t Table[QBinomial[n, 3, -10], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)

%o (Sage) [gaussian_binomial(n,3,-10) for n in range(3,14)] # _Zerinvary Lajos_, May 27 2009

%o (Magma) r:=3; q:=-10; [&*[(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