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Gaussian binomial coefficient [ n,5 ] for q = -3.
3

%I #29 Sep 08 2022 08:44:39

%S 1,-182,49777,-11662040,2869444942,-694405675964,168973319623174,

%T -41041673208656120,9974653139743515223,-2423717068608654822146,

%U 588973263031690760850991,-143119691677080990521708240

%N Gaussian binomial coefficient [ n,5 ] for q = -3.

%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="/A015306/b015306.txt">Table of n, a(n) for n = 5..200</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (-182,16653,428220,-4046679,-10746918,14348907)

%F G.f.: x^5/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)*(1-81*x)*(1+243*x)). - _R. J. Mathar_, Aug 03 2016

%F From _G. C. Greubel_, Sep 21 2019: (Start)

%F a(n) = (1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920.

%F E.g.f.: exp(-243*x)*(-1 +1830*exp(216*x) -44469*exp(240*x) +59049*exp(244 *x) -16470*exp(252*x) +61*exp(324*x))/1032762286080. (End)

%p seq((1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920, n=5..25); # _G. C. Greubel_, Sep 21 2019

%t Table[QBinomial[n, 5, -3], {n, 5, 20}] (* _Vincenzo Librandi_, Oct 29 2012 *)

%o (Sage) [gaussian_binomial(n,5,-3) for n in range(5,17)] # _Zerinvary Lajos_, May 27 2009

%o (PARI) a(n) = (1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920 \\ _G. C. Greubel_, Sep 21 2019

%o (Magma) [(1 - 61*(-3)^(n-4) + 610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) + 61*(-3)^(4*n-10) - (-3)^(5*n-10))/17489920: n in [5..25]]; // _G. C. Greubel_, Sep 21 2019

%o (GAP) List([5..25], n-> (1 -61*(-3)^(n-4) +610*(-3)^(2*n-7) - 610*(-3)^(3*n-9) +61*(-3)^(4*n-10) -(-3)^(5*n-10))/17489920); # _G. C. Greubel_, Sep 21 2019

%Y Gaussian binomial coefficients [n,5]: A015305 (q=-2), this sequence (q=-3), A015308 (q=-4), A015309 (q=-5), A015310 (q=-6), A015312 (q=-7), A015313 (q=-8), A015315 (q=-9), A015316 (q=-10), A015317 (q=-11), A015319 (q=-12), A015321 (q=-13).

%K sign,easy

%O 5,2

%A _Olivier GĂ©rard_, Dec 11 1999