A015273 Gaussian binomial coefficient [ n,3 ] for q=-6.

1, -185, 41107, -8838005, 1910490043, -412612541285, 89126228045659, -19251196169490725, 4158260859792814555, -898184256176675135525, 194007802557550502202331, -41905685236388916561230885

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 range(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