A015340 Gaussian binomial coefficient [ n,7 ] for q = -3.

1, -1640, 4035220, -8509702520, 18843459775162, -41041673208656120, 89881489830655851460, -196480936769813691291560, 429769342296322230713871283, -939857780045414554730512966640

Offset: 7

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 = 7..200
• Index entries for linear recurrences with constant coefficients, signature (-1640,1345620,314875080,-25929962838,-688631799960,6436058745780,17154979252920,-22876792454961).

Crossrefs

Gaussian binomial coefficient [n, k]_q for q = -3: A015251 (k = 2), A015268 (k = 3), A015288 (k = 4), A015306 (k = 5), A015324 (k = 6), this sequence (k = 7), A015357 (k = 8), A015375 (k = 9), A015388 (k = 10).

Programs

Mathematica

Table[QBinomial[n, 7, -3], {n, 7, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

Sage

[gaussian_binomial(n,7,-3) for n in range(7,17)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^7 / ( (x-1)*(27*x+1)*(81*x-1)*(729*x-1)*(9*x-1)*(2187*x+1)*(3*x+1)*(243*x+1) ). - R. J. Mathar, Sep 02 2016

G.f. with offset 0: exp(Sum_{n >= 1} A015518(8*n)/A015518(n) * (-x)^n/n) = 1 - 1640*x + 4035220*x^2 - .... - Peter Bala, Jun 29 2025