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

%I A015251 #30 Jun 30 2025 23:46:53
%S A015251 1,7,70,610,5551,49777,448540,4035220,36321901,326882347,2941985410,
%T A015251 26477735830,238300021051,2144698993717,19302294530680,
%U A015251 173720640014440,1563485792415001,14071372034879887
%N A015251 Gaussian binomial coefficient [ n,2 ] for q = -3.
%D A015251 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 A015251 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A015251 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A015251 G. C. Greubel, <a href="/A015251/b015251.txt">Table of n, a(n) for n = 2..500</a>
%H A015251 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,21,-27).
%F A015251 G.f.: x^2/[(1-x)(1+3x)(1-9x)].
%F A015251 a(n) = 10*a(n-1) - 9*a(n-2) + (-1)^n *3^(n-2), n >= 4. - _Vincenzo Librandi_, Mar 20 2011
%F A015251 a(n) = 7*a(n-1) + 21*a(n-2) - 27*a(n-3), n >= 3. - _Vincenzo Librandi_, Mar 20 2011
%F A015251 a(n) = (1/96)*(2*(-1)^n*3^n - 3 + 9^n). - _R. J. Mathar_, Mar 21 2011
%F A015251 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
%t A015251 Table[QBinomial[n, 2, -3], {n, 2, 25}] (* _G. C. Greubel_, Jul 30 2016 *)
%o A015251 (Sage) [gaussian_binomial(n,2,-3) for n in range(2,18)] # _Zerinvary Lajos_, May 28 2009
%o A015251 (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
%Y A015251 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).
%Y A015251 Cf. A015518.
%K A015251 nonn,easy
%O A015251 2,2
%A A015251 _Olivier Gérard_, Dec 11 1999