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Gaussian binomial coefficient [ n,3 ] for q = -13.
12

%I #31 Sep 08 2022 08:44:39

%S 1,-2040,4508570,-9900819720,21752862899691,-47790911017216080,

%T 104996653267533662740,-230677643550873536294640,

%U 506798783502833908602716981,-1113436927250681654567602842120

%N Gaussian binomial coefficient [ n,3 ] for q = -13.

%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="/A015286/b015286.txt">Table of n, a(n) for n = 3..200</a>

%H <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries related to Gaussian binomial coefficients</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2040,346970,4481880,-4826809)

%F a(n) = Product_{i=1..3} ((-13)^(n-i+1) - 1)/((-13)^i - 1). - _M. F. Hasler_, Nov 03 2012

%F G.f.: x^3 / ( (x-1)*(2197*x+1)*(13*x+1)*(169*x-1) ). - _R. J. Mathar_, Aug 03 2016

%e A015286(7) = 21752862899691 = A015303(7),

%e A015286(8) = -47790911017216080 = A015321(8),

%e A015286(9) = 104996653267533662740 = A015337(9). - _M. F. Hasler_, Nov 03 2012

%t QBinomial[Range[3,15],3,-13] (* _Harvey P. Dale_, Jun 21 2012 *)

%t Table[QBinomial[n, 3, -13], {n, 3, 20}] (* _Vincenzo Librandi_, Oct 28 2012 *)

%o (Sage) [gaussian_binomial(n,3,-13) for n in range(3,13)] # _Zerinvary Lajos_, May 27 2009

%o (PARI) A015286(n,r=3,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ _M. F. Hasler_, Nov 03 2012

%o (Magma) r:=3; q:=-13; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 02 2016

%Y Cf. Gaussian binomial coefficients [n,r] for q=-13: A015265 (r=2), A015303 (r=4), A015321 (r=5), A015337 (r=6), A015355 (r=7), A015370 (r=8), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - _M. F. Hasler_, Nov 03 2012

%Y Fourth row (r=3) or column (resp. diagonal) in A015129 (read as square array resp. triangle). - _M. F. Hasler_, Nov 03 2012

%K sign,easy

%O 3,2

%A _Olivier GĂ©rard_, Dec 11 1999