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Gaussian binomial coefficient [ n,8 ] for q=-6.
13

%I #21 Sep 08 2022 08:44:39

%S 1,1439671,2487182817955,4158260859792814555,

%T 6989674736616919292088715,11738459947705882553575280369515,

%U 19716527736890127515275338116221320235,33116077152651051199781730118147946460139435,55622326158904300663023790195853299389540017396395

%N Gaussian binomial coefficient [ n,8 ] for q=-6.

%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="/A015361/b015361.txt">Table of n, a(n) for n = 8..170</a>

%F a(n) = Product_{i=1..8} ((-6)^(n-i+1)-1)/((-6)^i-1). - _M. F. Hasler_, Nov 03 2012

%F G.f.: -x^8 / ( (x-1)*(279936*x+1)*(216*x+1)*(36*x-1)*(7776*x+1)*(1296*x-1)*(6*x+1)*(46656*x-1)*(1679616*x-1) ). - _R. J. Mathar_, Sep 02 2016

%t Table[QBinomial[n, 8, -6], {n, 8, 19}] (* _Vincenzo Librandi_, Nov 03 2012 *)

%o (Sage) [gaussian_binomial(n,8,-6) for n in range(8,15)] # _Zerinvary Lajos_, May 25 2009

%o (Magma) r:=8; q:=-6; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..18]]; // _Vincenzo Librandi_, Nov 03 2012

%o (PARI) A015361(n, r=8, q=-6)=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=-2..-13: A015356, A015357, A015359, A015360, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - _M. F. Hasler_, Nov 03 2012

%K nonn,easy

%O 8,2

%A _Olivier GĂ©rard_