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A006101 Gaussian binomial coefficient [ n,3 ] for q=3.

%I A006101 M5272 #45 Jul 14 2023 14:36:41
%S A006101 1,40,1210,33880,925771,25095280,678468820,18326727760,494894285941,
%T A006101 13362799477720,360801469802830,9741692640081640,263026177881648511,
%U A006101 7101711092201899360,191746238094034963240,5177148775980218655520,139783020078437440101481
%N A006101 Gaussian binomial coefficient [ n,3 ] for q=3.
%D A006101 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 A006101 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A006101 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006101 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006101 T. D. Noe, <a href="/A006101/b006101.txt">Table of n, a(n) for n=3..100</a>
%H A006101 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 A006101 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006101 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A006101 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40, -390, 1080, -729).
%F A006101 G.f.: z^3/((1-z)(1-3z)(1-9z)(1-27z)). _Simon Plouffe_ in his 1992 dissertation
%F A006101 a(n) = (27^n - 13*9^n + 39*3^n - 27)/11232. - _Mitch Harris_, Mar 23 2008
%t A006101 Table[QBinomial[n, 3, 3], {n, 3, 20}] (* _Vincenzo Librandi_, Nov 06 2016 *)
%o A006101 (Sage) [gaussian_binomial(n,3,3) for n in range(3,17)] # _Zerinvary Lajos_, May 25 2009
%o A006101 (Magma) r:=3; q:=3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Nov 06 2016
%K A006101 nonn
%O A006101 3,2
%A A006101 _N. J. A. Sloane_