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Gaussian binomial coefficient [ n,2 ] for q=4.
(Formerly M5115)
10

%I M5115 #52 Apr 13 2022 13:25:18

%S 1,21,357,5797,93093,1490853,23859109,381767589,6108368805,

%T 97734250405,1563749404581,25019996065701,400319959420837,

%U 6405119440211877,102481911401303973

%N Gaussian binomial coefficient [ n,2 ] for q=4.

%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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

%H Vincenzo Librandi, <a href="/A006105/b006105.txt">Table of n, a(n) for n = 2..200</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-84,64)

%F G.f.: x^2/((1-x)*(1-4*x)*(1-16*x)). [Multiplied by x^2 to match offset by _R. J. Mathar_, Mar 11 2009]

%F a(n) = (16^n - 5*4^n + 4)/180. - _Mitch Harris_, Mar 23 2008

%F a(n) = 5*a(n-1) -4*a(n-2) +16^(n-2), n>=4. - _Vincenzo Librandi_, Mar 20 2011

%p A006105:=-1/(z-1)/(4*z-1)/(16*z-1); # _Simon Plouffe_ in his 1992 dissertation, assuming offset zero

%t faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];

%t qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);

%t Table[qbin[n, 2, 4], {n, 2, 16}] (* _Jean-François Alcover_, Jul 21 2011 *)

%t CoefficientList[Series[1 / ((1 - x) (1 - 4 x) (1 - 16 x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 23 2013 *)

%t LinearRecurrence[{21,-84,64},{1,21,357},20] (* _Harvey P. Dale_, Feb 17 2020 *)

%o (Sage) [gaussian_binomial(n,2,4) for n in range(2,17)] # _Zerinvary Lajos_, May 28 2009

%K nonn

%O 2,2

%A _N. J. A. Sloane_