A015257 Gaussian binomial coefficient [ n,2 ] for q = -6.

1, 31, 1147, 41107, 1480963, 53308003, 1919128099, 69088371619, 2487182817955, 89538572808355, 3223388672928931, 116041991914472611, 4177511710786827427, 150390421577130906787, 5414055176843881927843, 194905986365976733701283

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 (31,186,-216).

Programs

Magma

I:=[1, 31, 1147]; [n le 3 select I[n] else 31*Self(n-1) + 186*Self(n-2) - 216*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 27 2012

Mathematica

Table[QBinomial[n, 2, -6], {n, 2, 20}] (* Vincenzo Librandi, Oct 27 2012 *)

Sage

[gaussian_binomial(n,2,-6) for n in range(2,17)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^2/((1-x)*(1+6*x)*(1-36*x)).

a(2) = 1, a(3) = 31, a(4) = 1147, a(n) = 31*a(n-1) + 186*a(n-2) - 216*a(n-3). - Vincenzo Librandi, Oct 27 2012