A015266 Gaussian binomial coefficient [ n,3 ] for q = -2.

1, -5, 55, -385, 3311, -25585, 208335, -1652145, 13275471, -105970865, 848699215, -6785865905, 54301841231, -434355079345, 3475079247695, -27799679551665, 222401254176591, -1779194762447025, 14233619183613775

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..1000
• Index entries for linear recurrences with constant coefficients, signature (-5, 30, 40, -64).

Crossrefs

Diagonal k=3 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Programs

Magma

[(1/81)*(24*4^n-6*(-2)^n+64*(-8)^n-1): n in [0..20]]; // Vincenzo Librandi, Aug 23 2011

Mathematica

Table[QBinomial[n, 2, -2], {n, 3, 25}] (* G. C. Greubel, Jul 31 2016 *)

Sage

[gaussian_binomial(n,3,-2) for n in range(3,22)] # Zerinvary Lajos, May 27 2009

Formula

From Paul Barry, Jul 12 2005: (Start)

G.f.: x^3/((1-2*x-8*x^2)*(1+7*x-8*x^2));

a(n) = -5*a(n-1) + 30*a(n-2) + 40*a(n-3) - 64*a(n-4);

a(n+3) = (-1)^n*J(n)*J(n+1)*J(n+2)/3, where J(n)=A001045(n). (End)

a(n) = T015109(n,3), where T015109 is the triangular array defined by A015109. - M. F. Hasler, Nov 04 2012