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A006097 Gaussian binomial coefficient [n, 4] for q = 2.

%I A006097 M5226 #61 Aug 15 2025 05:48:37
%S A006097 1,31,651,11811,200787,3309747,53743987,866251507,13910980083,
%T A006097 222984027123,3571013994483,57162391576563,914807651274739,
%U A006097 14638597687734259,234230965858250739,3747802679431278579,59965700687947706355,959458073589354016755,15351384078270441402355
%N A006097 Gaussian binomial coefficient [n, 4] for q = 2.
%D A006097 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 A006097 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A006097 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006097 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006097 T. D. Noe, <a href="/A006097/b006097.txt">Table of n, a(n) for n = 4..204</a>
%H A006097 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 A006097 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006097 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A006097 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (31,-310,1240,-1984,1024).
%F A006097 G.f.: x^4/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)).
%F A006097 a(n) = (2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160. - _Bruno Berselli_, Aug 29 2011
%F A006097 From _Peter Bala_, Jul 01 2025: (Start)
%F A006097 G.f. with an offset of 0: exp( Sum_{n >= 1} b(5*n)/b(n)*x^n/n ) = 1 + 31*x + 651*x^2 + ..., where b(n) = A000225(n) = 2^n - 1.
%F A006097 The following series telescope:
%F A006097 Sum_{n >= 4} 2^n/a(n) = 120/7; Sum_{n >= 4} 4^n/a(n) = 2078/7;
%F A006097 Sum_{n >= 4} 8^n/a(n) = 41280/7.
%F A006097 Sum_{n >= 4} 2^n/(a(n)*a(n+4)) = 40/499999;
%F A006097 Sum_{n >= 4} 2^n/(a(n)*a(n+4)*a(n+8)) = 40/6981154678721773;
%F A006097 Sum_{n >= 4} 2^n/(a(n)*a(n+4)*a(n+8)*a(n+12)) = 40/6387876185324781622646124392439. (End)
%p A006097 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 A006097 faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];
%t A006097 qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);
%t A006097 Table[qbin[n, 4, 2], {n, 4, 21}] (* _Jean-François Alcover_, Jul 21 2011 *)
%t A006097 QBinomial[Range[4,30],4,2] (* _Harvey P. Dale_, Dec 10 2012 *)
%o A006097 (Sage) [gaussian_binomial(n,4,2) for n in range(4,22)] # _Zerinvary Lajos_, May 24 2009
%o A006097 (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 A006097 (PARI) a(n)=(2^n-1)*(2^n-2)*(2^n-4)*(2^n-8)/20160 \\ _Charles R Greathouse IV_, Feb 19 2017
%Y A006097 Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), this sequence (k = 4), A006110 (k = 5), A022189 - A022195 (k = 6 thru 12).
%K A006097 nonn,easy,nice
%O A006097 4,2
%A A006097 _N. J. A. Sloane_