login
Gaussian binomial coefficient [ n,3 ] for q=2.
(Formerly M4982)
7

%I M4982 #52 Aug 22 2024 18:03:39

%S 1,15,155,1395,11811,97155,788035,6347715,50955971,408345795,

%T 3269560515,26167664835,209386049731,1675267338435,13402854502595,

%U 107225699266755,857817047249091,6862582190715075,54900840777134275,439207459223777475

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

%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="/A006096/b006096.txt">Table of n, a(n) for n=3..203</a>

%H Ronald Orozco López, <a href="https://arxiv.org/abs/2408.08943">Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function</a>, arXiv:2408.08943 [math.CO], 2024. See p. 13.

%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64).

%F G.f.: x^3/((1-x)(1-2x)(1-4x)(1-8x)).

%F (With a different offset) a(n)=(-1+7*2^n-14*4^n+8*8^n)/21. - _James R. Buddenhagen_, Dec 14 2003

%p seq((-1+7*2^n-14*4^n+8*8^n)/21,n=1..20);

%p A006096:=1/(z-1)/(8*z-1)/(2*z-1)/(4*z-1); # _Simon Plouffe_ in his 1992 dissertation with offset 0

%t Drop[CoefficientList[Series[x^3/((1 - x) (1 - 2 x) (1 - 4 x) (1 - 8 x)), {x, 0, 30}], x], 3]

%t QBinomial[Range[3,30],3,2] (* _Harvey P. Dale_, Jan 28 2013 *)

%o (Sage) [gaussian_binomial(n,3,2) for n in range(3,23)] # _Zerinvary Lajos_, May 24 2009

%o (Magma) r:=3; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // _Vincenzo Librandi_, Nov 06 2016

%K nonn,easy,nice

%O 3,2

%A _N. J. A. Sloane_