A015272 Gaussian binomial coefficient [ n,3 ] for q = -5.

1, -104, 13546, -1679704, 210302171, -26279294704, 3285123767796, -410635172794704, 51329529054158421, -6416187820400919704, 802023560334345174046, -100252942972187432169704

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 (-104,2730,13000,-15625).

Programs

Mathematica

Table[QBinomial[n, 3, -5], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)
LinearRecurrence[{-104,2730,13000,-15625},{1,-104,13546,-1679704},20] (* Harvey P. Dale, Apr 29 2022 *)

Sage

[gaussian_binomial(n,3,-5) for n in range(3,15)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+5*x)*(1-25*x)*(1+125*x)). - Bruno Berselli, Oct 29 2012

a(n) = (-1 + 21*5^(2n-3) + (-1)^n*5^(n-2)*(21-5^(2n-1)))/18144. - Bruno Berselli, Oct 29 2012