%I M5226 #51 Sep 08 2022 08:44:34
%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="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_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_