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A006100 Gaussian binomial coefficient [n, 2] for q = 3.

%I A006100 M4912 #47 Jul 05 2025 05:26:49
%S A006100 1,13,130,1210,11011,99463,896260,8069620,72636421,653757313,
%T A006100 5883904390,52955405230,476599444231,4289397389563,38604583680520,
%U A006100 347441274648040,3126971536402441
%N A006100 Gaussian binomial coefficient [n, 2] for q = 3.
%D A006100 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 A006100 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A006100 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006100 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006100 T. D. Noe, <a href="/A006100/b006100.txt">Table of n, a(n) for n=2..100</a>
%H A006100 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 A006100 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006100 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A006100 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13,-39,27).
%F A006100 G.f.: x^2/[(1-x)(1-3x)(1-9x)].
%F A006100 a(n) = (9^n - 4*3^n + 3)/48. - _Mitch Harris_, Mar 23 2008
%F A006100 a(n) = 4*a(n-1) -3*a(n-2) +9^(n-2), n>=4. - _Vincenzo Librandi_, Mar 20 2011
%F A006100 From _Peter Bala_, Jul 01 2025: (Start)
%F A006100 G.f. with an offset of 0: exp(Sum_{n >= 1} b(3*n)/b(n)*x^n/n) = 1 + 13*x + 130*x^2 + ..., where b(n) = 3^n - 1.
%F A006100 The following series telescope:
%F A006100 Sum_{n >= 2} 3^n/a(n) = 12; Sum_{n >= 2} 3^n/(a(n)*a(n+3)) = 129/16900;
%F A006100 Sum_{n >= 2} 9^n/(a(n)*a(n+3)) = 1227/16900;
%F A006100 Sum_{n >= 2} 3^n/(a(n)*a(n+3)*a(n+6)) = 156706257/18829431219368770. (End)
%p A006100 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 A006100:=-1/(z-1)/(3*z-1)/(9*z-1); # _Simon Plouffe_ in his 1992 dissertation with offset 0
%t A006100 f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
%t A006100 a[n_] := SymmetricPolynomial[2, t[n]]
%t A006100 Table[a[n], {n, 2, 32}]    (* A203243 *)
%t A006100 Table[a[n]/3, {n, 2, 32}]  (* A006100 *)
%o A006100 (Sage) [gaussian_binomial(n,2,3) for n in range(2,19)] # _Zerinvary Lajos_, May 25 2009
%Y A006100 Cf. A203243.
%K A006100 nonn
%O A006100 2,2
%A A006100 _N. J. A. Sloane_