A015276 Gaussian binomial coefficient [ n,3 ] for q = -8.

1, -455, 236665, -120935815, 61934287481, -31709385606535, 16235267484138105, -8312452980450674055, 4255976180162154314361, -2179059787976052939572615, 1115678612484825190455949945, -571227449525600988055816521095

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 (-455,29640,232960,-262144).

Programs

Magma

r:=3; q:=-8; [&*[(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, -8], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

Sage

[gaussian_binomial(n,3,-8) for n in range(3,15)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+8*x)*(1-64*x)*(1+512*x)). - Bruno Berselli, Oct 30 2012

a(n) = (-1 + 57*8^(2n-3) + (-1)^n*8^(n-2)*(57-8^(2n-1)))/290871. - Bruno Berselli, Oct 30 2012

a(n) = Product_{i=1..3} ((-8)^(n-i+1)-1)/((-8)^i-1) (by definition). - Vincenzo Librandi, Aug 02 2016