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A006096 Gaussian binomial coefficient [n, 3] for q = 2.

%I A006096 M4982 #70 Jul 07 2025 00:31:24
%S A006096 1,15,155,1395,11811,97155,788035,6347715,50955971,408345795,
%T A006096 3269560515,26167664835,209386049731,1675267338435,13402854502595,
%U A006096 107225699266755,857817047249091,6862582190715075,54900840777134275,439207459223777475
%N A006096 Gaussian binomial coefficient [n, 3] for q = 2.
%C A006096 42*a(n) is a maximum number of intercalates in a Latin square of order 2^n-1 (see A092237). - _Eduard I. Vatutin_, Apr 24 2025
%D A006096 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 A006096 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A006096 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006096 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006096 T. D. Noe, <a href="/A006096/b006096.txt">Table of n, a(n) for n=3..203</a>
%H A006096 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 A006096 Ronald Orozco López, <a href="https://arxiv.org/abs/2501.11490">Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials</a>, arXiv:2501.11490 [math.CO], 2025. See p. 10.
%H A006096 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 A006096 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006096 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A006096 Eduard I. Vatutin, <a href="https://vk.com/wall162891802_3015">About interconnection between maximum number of intercalates in Latin squares of order N=2^n-1 and Gaussian binomial coefficients [n,3] for q=2</a> (in Russian).
%H A006096 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64).
%F A006096 G.f.: x^3/((1-x)(1-2x)(1-4x)(1-8x)).
%F A006096 (With a different offset) a(n) = (-1+7*2^n-14*4^n+8*8^n)/21. - _James R. Buddenhagen_, Dec 14 2003
%F A006096 From _Peter Bala_, Jul 01 2025: (Start)
%F A006096 a(n) = (q^n - 1)*(q^(n-1) - 1)*(q^(n-2) - 1)/((q^3 - 1)*(q^2 - 1)*(q - 1)) at q = 2.
%F A006096 G.f. with an offset of 0: exp( Sum_{n >= 1} b(4*n)/b(n)*x^n/n ) = 1 + 15*x + 155*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.
%F A006096 The following series telescope:
%F A006096 Sum_{n >= 3} 2^n/(a(n)*a(n+3)) = 420/72075;
%F A006096 Sum_{n >= 3} 4^n/(a(n)*a(n+3)) = 3416/72075;
%F A006096 Sum_{n >= 3} 8^n/(a(n)*a(n+3)) = 28296/72075;
%F A006096 Sum_{n >= 3} 16^n/(a(n)*a(n+3)) = 244748/72075;
%F A006096 Sum_{n >= 3} 32^n/(a(n)*a(n+3)) = 2415315/72075. (End)
%p A006096 seq((-1+7*2^n-14*4^n+8*8^n)/21,n=1..20);
%p A006096 A006096:=1/(z-1)/(8*z-1)/(2*z-1)/(4*z-1); # _Simon Plouffe_ in his 1992 dissertation with offset 0
%t A006096 Drop[CoefficientList[Series[x^3/((1 - x) (1 - 2 x) (1 - 4 x) (1 - 8 x)), {x, 0, 30}], x], 3]
%t A006096 QBinomial[Range[3,30],3,2] (* _Harvey P. Dale_, Jan 28 2013 *)
%o A006096 (Sage) [gaussian_binomial(n,3,2) for n in range(3,23)] # _Zerinvary Lajos_, May 24 2009
%o A006096 (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
%Y A006096 Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), this sequence (k = 3), A006097 (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).
%Y A006096 Cf. A092237.
%K A006096 nonn,easy,nice
%O A006096 3,2
%A A006096 _N. J. A. Sloane_