cp's OEIS Frontend

A006106 Gaussian binomial coefficient [ n,3 ] for q = 4.

%I A006106 M5360 #43 Jul 14 2023 14:39:05
%S A006106 1,85,5797,376805,24208613,1550842085,99277752549,6354157930725,
%T A006106 406672215935205,26027119554103525,1665737215212030181,
%U A006106 106607206793565997285,6822861635108183247077,436663151052043168024805,27946441769812674154891493
%N A006106 Gaussian binomial coefficient [ n,3 ] for q = 4.
%D A006106 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 A006106 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A006106 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006106 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006106 Vincenzo Librandi, <a href="/A006106/b006106.txt">Table of n, a(n) for n = 3..200</a>
%H A006106 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 A006106 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006106 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A006106 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (85, -1428, 5440, -4096).
%F A006106 G.f.: x^3/((1-x)*(1-4*x)*(1-16*x)*(1-64*x)). - _Simon Plouffe_ in his 1992 dissertation
%F A006106 a(n) = Product_{i=1..3} (4^(n-i+1)-1)/(4^i-1), by definition. - _Vincenzo Librandi_, Aug 07 2016
%t A006106 Table[QBinomial[n, 3, 4], {n, 3, 20}] (* _Vincenzo Librandi_, Aug 07 2016 *)
%o A006106 (Sage) [gaussian_binomial(n,3,4) for n in range(3,15)] # _Zerinvary Lajos_, May 27 2009
%o A006106 (Magma) r:=3; q:=4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 07 2016
%K A006106 nonn,easy
%O A006106 3,2
%A A006106 _N. J. A. Sloane_