%I M4912 #43 Apr 13 2022 13:25:18

%S 1,13,130,1210,11011,99463,896260,8069620,72636421,653757313,

%T 5883904390,52955405230,476599444231,4289397389563,38604583680520,

%U 347441274648040,3126971536402441

%N Gaussian binomial coefficient [ n,2 ] for q=3.

%D 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 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

%H T. D. Noe, <a href="/A006100/b006100.txt">Table of n, a(n) for n=2..100</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13,-39,27).

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

%F a(n) = (9^n - 4*3^n + 3)/48. - _Mitch Harris_, Mar 23 2008

%F a(n) = 4*a(n-1) -3*a(n-2) +9^(n-2), n>=4. - _Vincenzo Librandi_, Mar 20 2011

%p a:=n->sum((9^(n-j)-3^(n-j))/6,j=0..n): seq(a(n), n=1..17); # _Zerinvary Lajos_, Jan 15 2007

%p A006100:=-1/(z-1)/(3*z-1)/(9*z-1); # _Simon Plouffe_ in his 1992 dissertation with offset 0

%t f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[2, t[n]]

%t Table[a[n], {n, 2, 32}] (* A203243 *)

%t Table[a[n]/3, {n, 2, 32}] (* A006100 *)

%o (Sage) [gaussian_binomial(n,2,3) for n in range(2,19)] # _Zerinvary Lajos_, May 25 2009

%Y Cf. A203243.

%K nonn

%O 2,2

%A _N. J. A. Sloane_