A015271 Gaussian binomial coefficient [ n,3 ] for q = -4.

1, -51, 3485, -219555, 14107485, -901984419, 57741320029, -3695215419555, 236497451900765, -15135778281070755, 968690748238618461, -61996192875273494691, 3967756584209486471005, -253936417546335462858915

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 (-51,884,3264,-4096).

Programs

Mathematica

Table[QBinomial[n, 3, -4], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

Sage

[gaussian_binomial(n,3,-4) for n in range(3,17)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+4*x)*(1-16*x)*(1+64*x)). - Bruno Berselli, Oct 29 2012

a(n) = (-1 + 13*2^(4n-6) + (-1)^n*4^(n-2)*(13-2^(4n-2)))/4875. - Bruno Berselli, Oct 29 2012

a(n) = -51*a(n-1)+884*a(n-2)+3264*a(n-3)-4096*a(n-4). - Wesley Ivan Hurt, Sep 04 2022