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%I #22 Sep 08 2022 08:44:39
%S 1,52429,3665049245,236497451900765,15559876852907031645,
%T 1018737244037427165087837,66780267552779682073190144093,
%U 4376244513647234644625387176712285,286805936690898816904813999400193022045
%N Gaussian binomial coefficient [ n,8 ] 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 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H Vincenzo Librandi, <a href="/A015359/b015359.txt">Table of n, a(n) for n = 8..200</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (52429,916249204,-3695444481856,-3798047410999296,972300137215819776,61999270288106192896,-1007426653738504290304,-3777907597814523756544,4722366482869645213696).
%F a(n) = Product_{i=1..8} ((-4)^(n-i+1)-1)/((-4)^i-1). - _M. F. Hasler_, Nov 03 2012
%F G.f.: -x^8 / ( (x-1)*(16384*x+1)*(4096*x-1)*(256*x-1)*(65536*x-1)*(64*x+1)*(4*x+1)*(16*x-1)*(1024*x+1) ). - _R. J. Mathar_, Sep 02 2016
%t Table[QBinomial[n, 8, -4], {n, 8, 20}] (* _Vincenzo Librandi_, Nov 02 2012 *)
%o (Sage) [gaussian_binomial(n,8,-4) for n in range(8,16)] # _Zerinvary Lajos_, May 25 2009
%o (Magma) r:=8; q:=-4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // _Vincenzo Librandi_, Nov 03 2012
%o (PARI) A015359(n,r=8,q=-4)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ _M. F. Hasler_, Nov 03 2012
%Y Cf. A015356, A015357, A015360, A015361, A015363, A015364, A015365, A015367 A015368, A015369, A015370 (r=8, q=-2..-13). q=-4 integers/coefficients: A014985 (r=1), A015253 (r=2), A015271 (r=3), A015289 (r=4), A015308 (r=5), A015326 (r=6), A015341 (r=7), A015376 (r=9), A015390 (r=10), A015408 (r=11), A015425 (r=12). - _M. F. Hasler_, Nov 03 2012
%K nonn,easy
%O 8,2
%A _Olivier Gérard_
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