A015278 Gaussian binomial coefficient [ n,3 ] for q = -10.

1, -909, 918191, -917272809, 917364637191, -917355454462809, 917356372736537191, -917356280909173462809, 917356290091909926537191, -917356289173636281073462809, 917356289265463645628926537191

Offset: 3

Author

Olivier Gérard, Dec 11 1999

Keywords

• sign
• easy
• changed

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
• Umesh Shankar, Log-concavity of rows of triangular arrays satisfying a certain super-recurrence, arXiv:2508.12467 [math.CO], 2025. See p. 4.
• Index entries for linear recurrences with constant coefficients, signature (-909,91910,909000,-1000000).

Programs

Magma

r:=3; q:=-10; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 02 2016

Mathematica

Table[QBinomial[n, 3, -10], {n, 3, 20}] (* Vincenzo Librandi, Oct 28 2012 *)

Sage

[gaussian_binomial(n,3,-10) for n in range(3,14)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+10*x)*(1-100*x)*(1+1000*x)). - Bruno Berselli, Oct 30 2012

a(n) = (-1 + 91*10^(2n-3) + (-1)^n*10^(n-2)*(91-10^(2n-1)))/1090089. - Bruno Berselli, Oct 30 2012

a(n) = Product_{i=1..3} ((-10)^(n-i+1)-1)/((-10)^i-1) (by definition). - Vincenzo Librandi, Aug 02 2016