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Gaussian binomial coefficient [ n,2 ] for q = -7.
3

%I #30 Sep 08 2022 08:44:39

%S 1,43,2150,105050,5149551,252313293,12363454300,605808540100,

%T 29684623509101,1454546516636543,71272779562356450,

%U 3492366196825305150,171125943656551078651,8385171239086224969793,410873390715818468708600,20132796145070950850400200

%N Gaussian binomial coefficient [ n,2 ] for q = -7.

%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="/A015258/b015258.txt">Table of n, a(n) for n = 2..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (43,301,-343).

%F G.f.: x^2/((1-x)*(1+7x)*(1-49x)).

%F a(n) = (6*(-7)^n - 7 +49^n)/2688. - _R. J. Mathar_, May 25 2011

%F a(n) = 43*a(n-1) + 301*a(n-2) - 343*a(n-3), n >= 5. - _Harvey P. Dale_, May 25 2011

%t CoefficientList[Series[1/((1-x)(1+7x)(1-49x)),{x,0,20}],x] (* or *) LinearRecurrence[{43,301,-343},{1,43,2150},20] (* _Harvey P. Dale_, May 25 2011 *)

%t Table[QBinomial[n, 2, -7], {n, 2, 20}] (* _Vincenzo Librandi_, Oct 27 2012 *)

%o (Sage) [gaussian_binomial(n,2,-7) for n in range(2,16)] # _Zerinvary Lajos_, May 27 2009

%o (Magma) I:=[1, 43, 2150]; [n le 3 select I[n] else 43*Self(n-1) + 301*Self(n-2) - 343*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Oct 27 2012

%K nonn,easy

%O 2,2

%A _Olivier GĂ©rard_, Dec 11 1999