A015277 Gaussian binomial coefficient [ n,3 ] for q = -9.

1, -656, 484210, -352504880, 257015284435, -187360965026144, 136586400868021924, -99571465386311288480, 72587599955185580267365, -52916360230556551635386480, 38576026619154398792076180886

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 (-656,53874,478224,-531441).

Programs

Magma

r:=3; q:=-9; [&*[(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, -9], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)
LinearRecurrence[{-656,53874,478224,-531441},{1,-656,484210,-352504880},20] (* Harvey P. Dale, Feb 10 2015 *)

Sage

[gaussian_binomial(n,3,-9) for n in range(3,14)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+9*x)*(1-81*x)*(1+729*x)). - Bruno Berselli, Oct 30 2012

a(n) = (-1 + 73*3^(4n-6) + (-1)^n*3^(2n-4)*(73-3^(4n-2)))/584000. - Bruno Berselli, Oct 30 2012

a(n) = product(((-9)^(n-i+1)-1)/((-9)^i-1), i=1..3) (by definition). - Vincenzo Librandi, Aug 02 2016