A015259 Gaussian binomial coefficient [ n,2 ] for q = -8.

1, 57, 3705, 236665, 15150201, 969583737, 62053592185, 3971428035705, 254171409198201, 16266970069380217, 1041086085394771065, 66629509457629850745, 4264288605349394427001, 272914470741872571493497

Offset: 2

Author

Olivier Gérard, Dec 11 1999

Keywords

• nonn
• 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 = 2..200
• Index entries for linear recurrences with constant coefficients, signature (57,456,-512).

Programs

Magma

I:=[1, 57, 3705]; [n le 3 select I[n] else 57*Self(n-1)+456*Self(n-2)-512*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 27 2012

Mathematica

Table[QBinomial[n, 2, -8], {n, 2, 20}] (* Vincenzo Librandi, Oct 27 2012 *)

Sage

[gaussian_binomial(n,2,-8) for n in range(2,16)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^2/((1-x)*(1+8*x)*(1-64*x)).

a(2) = 1, a(3) = 57, a(4) = 3705, a(n) = 57*a(n-1) + 456*a(n-2) - 512*a(n-3). - Vincenzo Librandi, Oct 27 2012