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

%I A015249 #40 Apr 25 2025 10:28:36
%S A015249 1,3,15,55,231,903,3655,14535,58311,232903,932295,3727815,14913991,
%T A015249 59650503,238612935,954429895,3817763271,15270965703,61084037575,
%U A015249 244335800775,977343902151,3909374210503,15637499638215
%N A015249 Gaussian binomial coefficient [ n,2 ] for q = -2.
%D A015249 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.
%D A015249 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A015249 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A015249 G. C. Greubel, <a href="/A015249/b015249.txt">Table of n, a(n) for n = 2..500</a>
%H A015249 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8)
%F A015249 G.f.: x^2/((1-x)*(1+2*x)*(1-4*x)).
%F A015249 From _Vincenzo Librandi_, Mar 20 2011: (Start)
%F A015249 a(n) = 5*a(n-1) - 4*a(n-2) + (-1)^n *2^(n-2), n >= 4.
%F A015249 a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), n >= 3. (End)
%F A015249 a(n) = (1/18)*(4^n - 2 + (-1)^n*2^n). - _R. J. Mathar_, Mar 21 2011
%F A015249 E.g.f.: 2*exp(x)*sinh(3*x/2)^2/9. - _Stefano Spezia_, Apr 25 2025
%t A015249 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 *)
%t A015249 Table[QBinomial[n, 2, -2], {n, 2, 25}] (* _G. C. Greubel_, Jul 30 2016 *)
%o A015249 (Sage) [gaussian_binomial(n,2,-2) for n in range(2,25)] # _Zerinvary Lajos_, May 28 2009
%o A015249 (PARI) a(n)=(4^n - 2 + (-1)^n*2^n)/18 \\ _Charles R Greathouse IV_, Jul 30 2016
%o A015249 (Python)
%o A015249 def A015249(n): return ((m:=1<<n)|1)//3*((m>>1|1)//3) # _Chai Wah Wu_, Apr 25 2025
%Y A015249 Except for initial terms, same as A084152 and A084175.
%K A015249 nonn,easy
%O A015249 2,2
%A A015249 _Olivier Gérard_, Dec 11 1999