A015249 Gaussian binomial coefficient [ n,2 ] for q = -2.

1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575, 244335800775, 977343902151, 3909374210503, 15637499638215

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 (3,6,-8)

Crossrefs

Except for initial terms, same as A084152 and A084175.

Programs

Mathematica

Join[{a=1,b=3},Table[c=2*b+8*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011 *)
Table[QBinomial[n, 2, -2], {n, 2, 25}] (* G. C. Greubel, Jul 30 2016 *)

PARI

a(n)=(4^n - 2 + (-1)^n*2^n)/18 \\ Charles R Greathouse IV, Jul 30 2016

Python

def A015249(n): return ((m:=1<>1|1)//3) # Chai Wah Wu, Apr 25 2025

Sage

[gaussian_binomial(n,2,-2) for n in range(2,25)] # Zerinvary Lajos, May 28 2009

Formula

G.f.: x^2/((1-x)*(1+2*x)*(1-4*x)).

From Vincenzo Librandi, Mar 20 2011: (Start)

a(n) = 5*a(n-1) - 4*a(n-2) + (-1)^n *2^(n-2), n >= 4.

a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), n >= 3. (End)

a(n) = (1/18)*(4^n - 2 + (-1)^n*2^n). - R. J. Mathar, Mar 21 2011

E.g.f.: 2*exp(x)*sinh(3*x/2)^2/9. - Stefano Spezia, Apr 25 2025