A015251 Gaussian binomial coefficient [ n,2 ] for q = -3.

1, 7, 70, 610, 5551, 49777, 448540, 4035220, 36321901, 326882347, 2941985410, 26477735830, 238300021051, 2144698993717, 19302294530680, 173720640014440, 1563485792415001, 14071372034879887

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

• G. C. Greubel, Table of n, a(n) for n = 2..500
• Index entries for linear recurrences with constant coefficients, signature (7,21,-27).

Crossrefs

Gaussian binomial coefficient [n, k]_q for q = -3: A015268 (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

Mathematica

Table[QBinomial[n, 2, -3], {n, 2, 25}] (* G. C. Greubel, Jul 30 2016 *)

PARI

a(n)=([0,1,0; 0,0,1; -27,21,7]^(n-2)*[1;7;70])[1,1] \\ Charles R Greathouse IV, Jul 30 2016

Sage

[gaussian_binomial(n,2,-3) for n in range(2,18)] # Zerinvary Lajos, May 28 2009

Formula

G.f.: x^2/[(1-x)(1+3x)(1-9x)].

a(n) = 10*a(n-1) - 9*a(n-2) + (-1)^n *3^(n-2), n >= 4. - Vincenzo Librandi, Mar 20 2011

a(n) = 7*a(n-1) + 21*a(n-2) - 27*a(n-3), n >= 3. - Vincenzo Librandi, Mar 20 2011

a(n) = (1/96)*(2*(-1)^n*3^n - 3 + 9^n). - R. J. Mathar, Mar 21 2011

G.f. with offset 0: exp(Sum_{n >= 1} A015518(3*n)/A015518(n) * x^n/n) = 1 + 7*x + 70*x^2 + .... - Peter Bala, Jun 29 2025