A015258 Gaussian binomial coefficient [ n,2 ] for q = -7.

1, 43, 2150, 105050, 5149551, 252313293, 12363454300, 605808540100, 29684623509101, 1454546516636543, 71272779562356450, 3492366196825305150, 171125943656551078651, 8385171239086224969793, 410873390715818468708600, 20132796145070950850400200

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 (43,301,-343).

Programs

Magma

I:=[1, 43, 2150]; [n le 3 select I[n] else 43*Self(n-1) + 301*Self(n-2) - 343*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 27 2012

Mathematica

CoefficientList[Series[1/((1-x)(1+7x)(1-49x)),{x,0,20}],x] (* or *) LinearRecurrence[{43,301,-343},{1,43,2150},20] (* Harvey P. Dale, May 25 2011 *)
Table[QBinomial[n, 2, -7], {n, 2, 20}] (* Vincenzo Librandi, Oct 27 2012 *)

Sage

[gaussian_binomial(n,2,-7) for n in range(2,16)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^2/((1-x)*(1+7x)*(1-49x)).

a(n) = (6*(-7)^n - 7 +49^n)/2688. - R. J. Mathar, May 25 2011

a(n) = 43*a(n-1) + 301*a(n-2) - 343*a(n-3), n >= 5. - Harvey P. Dale, May 25 2011