A015281 Gaussian binomial coefficient [ n,3 ] for q = -12.

1, -1595, 2775445, -4793193515, 8283038077141, -14313032243145515, 24732928003956401365, -42738498397393357626155, 73852125402551558141191381, -127616472670861852065241422635

Offset: 3

Author

Olivier Gérard, Dec 11 1999

Keywords

• sign
• easy

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 (-1595,231420,2756160,-2985984).

Programs

Magma

r:=3; q:=-12; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Mathematica

Table[QBinomial[n, 3, -12], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

Sage

[gaussian_binomial(n,3,-12) for n in range(3,13)] # Zerinvary Lajos, May 27 2009

Formula

a(n) = Product_{i=1..3} ((-12)^(n-i+1)-1)/((-12)^i-1) (by definition). - Vincenzo Librandi, Aug 02 2016

G.f.: x^3 / ( (x-1)*(12*x+1)*(1728*x+1)*(144*x-1) ). - R. J. Mathar, Aug 03 2016