A015268 Gaussian binomial coefficient [ n,3 ] for q = -3.

1, -20, 610, -15860, 433771, -11662040, 315323620, -8509702520, 229798289941, -6204226946060, 167517069529030, -4522934399547980, 122119467087816511, -3297223466672052080, 89025052902439936840, -2403676254645238280240

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..200
• Index entries for linear recurrences with constant coefficients, signature (-20,210,540,-729).

Crossrefs

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), this sequence (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), A015340 (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Cf. A015518.

Programs

Magma

[(-1+7*3^(2*n-3)+(-1)^n*3^(n-2)*(7-3^(2*n-1)))/896: n in [3..18]]; // Bruno Berselli, Oct 29 2012

Mathematica

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

Maxima

makelist(coeff(taylor(1/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)), x, 0, n), x, n), n, 0, 15); /* Bruno Berselli, Oct 29 2012 */

SageMath

[gaussian_binomial(n,3,-3) for n in range(3,19)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1+3*x)*(1-9*x)*(1+27*x)). - Bruno Berselli, Oct 29 2012

a(n) = (-1 + 7*3^(2n-3) + (-1)^n*3^(n-2)*(7-3^(2n-1)))/896. - Bruno Berselli, Oct 29 2012

G.f.: x^3 * exp(Sum_{n >= 1} A015518(4*n)/A015518(n) * (-x)^n/n) = x^3 * (1 - 20*x + 610*x^2 - ...). - Peter Bala, Jun 29 2025