A015275 Gaussian binomial coefficient [ n,3 ] for q = -7.

1, -300, 105050, -35927100, 12328144851, -4228301370600, 1450319733570100, -497459062806004200, 170628488227082949701, -58525570007342935110900, 20074270583791406305395150, -6885474806748086165925231300

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 (-300,15050,102900,-117649).

Programs

Magma

r:=3; q:=-7; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Mathematica

QBinomial[Range[3,20],3,-7] (* Harvey P. Dale, Apr 09 2012 *)
Table[QBinomial[n, 3, -7], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

Sage

[gaussian_binomial(n,3,-7) for n in range(3,15)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+7*x)*(1-49*x)*(1+343*x)). - Bruno Berselli, Oct 30 2012

a(n) = (-1 + 43*7^(2n-3) + (-1)^n*7^(n-2)*(43-7^(2n-1)))/132096. - Bruno Berselli, Oct 30 2012