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%I #27 Sep 08 2022 08:44:39
%S 1,171,58311,13275471,3624203583,899790907743,233988483199263,
%T 59438516325245343,15275698695588053151,3902985682508407194271,
%U 1000137219716325891620511,255910660218571393553843871
%N Gaussian binomial coefficient [ n,8 ] for q=-2.
%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="/A015356/b015356.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 (171,29070,-1666680,-56000448,896007168,6826721280,-30482104320,-45902462976,68719476736).
%F a(n) = Product_{i=1..8} ((-2)^(n-i+1)-1)/((-2)^i-1). - _M. F. Hasler_, Nov 03 2012
%F G.f.: -x^8 / ( (x-1)*(64*x-1)*(128*x+1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - _R. J. Mathar_, Sep 02 2016
%t Table[QBinomial[n, 8, -2], {n, 8, 20}] (* _Vincenzo Librandi_, Nov 02 2012 *)
%o (Sage) [gaussian_binomial(n,8,-2) for n in range(8,20)] # _Zerinvary Lajos_, May 25 2009
%o (Magma) r:=8; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // _Vincenzo Librandi_, Nov 02 2012
%o (PARI) A015356(n, r=8, q=-2)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ _M. F. Hasler_, Nov 03 2012
%Y Cf. Gaussian binomial coefficients [n,8] for q=-3..-13: A015357, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - _M. F. Hasler_, Nov 03 2012
%Y Diagonal k=8 of the triangular array A015109. See there for further references and programs. - _M. F. Hasler_, Nov 04 2012
%K nonn,easy
%O 8,2
%A _Olivier Gérard_
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