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A015278 Gaussian binomial coefficient [ n,3 ] for q = -10.

%I A015278 #34 Aug 22 2025 11:05:40
%S A015278 1,-909,918191,-917272809,917364637191,-917355454462809,
%T A015278 917356372736537191,-917356280909173462809,917356290091909926537191,
%U A015278 -917356289173636281073462809,917356289265463645628926537191
%N A015278 Gaussian binomial coefficient [ n,3 ] for q = -10.
%D A015278 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 A015278 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A015278 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A015278 Vincenzo Librandi, <a href="/A015278/b015278.txt">Table of n, a(n) for n = 3..200</a>
%H A015278 Umesh Shankar, <a href="https://arxiv.org/abs/2508.12467">Log-concavity of rows of triangular arrays satisfying a certain super-recurrence</a>, arXiv:2508.12467 [math.CO], 2025. See p. 4.
%H A015278 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-909,91910,909000,-1000000).
%F A015278 G.f.: x^3/((1-x)*(1+10*x)*(1-100*x)*(1+1000*x)). - _Bruno Berselli_, Oct 30 2012
%F A015278 a(n) = (-1 + 91*10^(2n-3) + (-1)^n*10^(n-2)*(91-10^(2n-1)))/1090089. - _Bruno Berselli_, Oct 30 2012
%F A015278 a(n) = Product_{i=1..3} ((-10)^(n-i+1)-1)/((-10)^i-1) (by definition). - _Vincenzo Librandi_, Aug 02 2016
%t A015278 Table[QBinomial[n, 3, -10], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)
%o A015278 (Sage) [gaussian_binomial(n,3,-10) for n in range(3,14)] # _Zerinvary Lajos_, May 27 2009
%o A015278 (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 A015278 sign,easy,changed
%O A015278 3,2
%A A015278 _Olivier Gérard_, Dec 11 1999