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

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

Offset: 3

Author

Olivier Gérard, Dec 11 1999

Keywords

• sign
• easy

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).

Programs

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

Mathematica

Table[QBinomial[n, 3, -6], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

Sage

[gaussian_binomial(n,3,-6) for n in range(3,15)] # Zerinvary Lajos, May 27 2009

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