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A006097 Gaussian binomial coefficient [ n,4 ] for q=2.
(Formerly M5226)
5

%I M5226

%S 1,31,651,11811,200787,3309747,53743987,866251507,13910980083,

%T 222984027123,3571013994483,57162391576563,914807651274739,

%U 14638597687734259,234230965858250739,3747802679431278579,59965700687947706355,959458073589354016755

%N Gaussian binomial coefficient [ n,4 ] 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="/A006097/b006097.txt">Table of n, a(n) for n = 4..204</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 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_05">Index entries for linear recurrences with constant coefficients</a>, signature (31,-310,1240,-1984,1024).

%F G.f.: x^4/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)).

%F a(n) = (2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160. - _Bruno Berselli_, Aug 29 2011

%p A006097:=-1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1); # _Simon Plouffe_ in his 1992 dissertation with offset 0

%t faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];

%t qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);

%t Table[qbin[n, 4, 2], {n, 4, 21}] (* _Jean-François Alcover_, Jul 21 2011 *)

%t QBinomial[Range[4,30],4,2] (* _Harvey P. Dale_, Dec 10 2012 *)

%o (Sage) [gaussian_binomial(n,4,2) for n in range(4,22)] # _Zerinvary Lajos_, May 24 2009

%o (MAGMA) r:=4; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // _Vincenzo Librandi_, Nov 06 2016

%o (PARI) a(n)=(2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160 \\ _Charles R Greathouse IV_, Feb 19 2017

%K nonn,easy,nice

%O 4,2

%A _N. J. A. Sloane_

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Last modified August 7 09:16 EDT 2020. Contains 336274 sequences. (Running on oeis4.)