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 A015273 Gaussian binomial coefficient [ n,3 ] for q=-6. 2
 1, -185, 41107, -8838005, 1910490043, -412612541285, 89126228045659, -19251196169490725, 4158260859792814555, -898184256176675135525, 194007802557550502202331, -41905685236388916561230885 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 COMMENTS From Bruno Berselli, Oct 30 2012: (Start) More generally, for sequences of the type "Gaussian binomial coefficient [n,3] for q=-m", we have: a(n) = (1-(-m)^n)*(1-(-m)^(n-1))*(1-(-m)^(n-2))/((1+m)*(1-m^2)*(1+m^3)) = (-1+(1-m+m^2)*m^(2n-3)+(-1)^n*m^(n-2)*(1-m+m^2-m^(2n-1)))/(-1-m+m^2-m^4+m^5+m^6), G.f.: x^3/((1-x)*(1+m*x)*(1-m^2*x)*(1+m^3*x)). (End) REFERENCES 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. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99. M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. LINKS Vincenzo Librandi, Table of n, a(n) for n = 3..200 Index entries for linear recurrences with constant coefficients, signature (-185,6882,39960,-46656). FORMULA G.f.: x^3/((1-x)*(1+6*x)*(1-36*x)*(1+216*x)). - Bruno Berselli, Oct 30 2012 a(n) = (-1+31*6^(2n-3)+(-1)^n*6^(n-2)*(31-6^(2n-1)))/53165. - Bruno Berselli, Oct 30 2012 MATHEMATICA Table[QBinomial[n, 3, -6], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *) PROG (Sage) [gaussian_binomial(n, 3, -6) for n in xrange(3, 15)] # Zerinvary Lajos, May 27 2009 (MAGMA) I:=[1, -185, 41107, -8838005]; [n le 4 select I[n] else -185*Self(n-1)+6882*Self(n-2)+39960*Self(n-3)-46656*Self(n-4): n in [1..13]]; // Vincenzo Librandi, Oct 29 2012 CROSSREFS Sequence in context: A189181 A203392 A197646 * A214605 A186398 A124207 Adjacent sequences:  A015270 A015271 A015272 * A015274 A015275 A015276 KEYWORD sign,easy AUTHOR Olivier Gérard, Dec 11 1999 STATUS approved

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