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A006110 Gaussian binomial coefficient [n, 5] for q = 2.

%I A006110 M5327 #58 Jul 05 2025 05:20:39
%S A006110 1,63,2667,97155,3309747,109221651,3548836819,114429029715,
%T A006110 3675639930963,117843461817939,3774561792168531,120843139740969555,
%U A006110 3867895279362300499,123787287537281350227,3961427432158861458003,126769425631762997934675,4056681585917103881615955,129814770207420913565727315
%N A006110 Gaussian binomial coefficient [n, 5] for q = 2.
%D A006110 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 A006110 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
%D A006110 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006110 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006110 Vincenzo Librandi, <a href="/A006110/b006110.txt">Table of n, a(n) for n = 5..200</a>
%H A006110 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 A006110 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006110 M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%H A006110 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (63,-1302,11160,-41664,64512,-32768).
%F A006110 a(n+4) = (1024*32^n-1984*16^n+1240*8^n-310*4^n+31*2^n-1)/9765. - _James R. Buddenhagen_, Dec 14 2003
%F A006110 G.f.: x^5/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x)*(1-32*x)). - _Vincenzo Librandi_, Aug 07 2016
%F A006110 a(n) = Product_{i=1..5} (2^(n-i+1)-1)/(2^i-1), by definition. - _Vincenzo Librandi_, Aug 06 2016
%F A006110 a(n) = (2^n-16)*(2^n-8)*(2^n-4)*(2^n-2)*(2^n-1)/9999360. - _Robert Israel_, Feb 01 2018
%F A006110 G.f. with an offset of 0: exp( Sum_{n >= 1} b(6*n)/b(n)*x^n/n ) = 1 + 63*x + 2667*x^2 + ..., where b(n) = A000225(n) = 2^n - 1. - _Peter Bala_, Jul 01 2025
%p A006110 seq((1024*32^n-1984*16^n+1240*8^n-310*4^n+31*2^n-1)/9765,n=1..20);
%p A006110 A006110:=1/(z-1)/(4*z-1)/(2*z-1)/(8*z-1)/(16*z-1)/(32*z-1); # _Simon Plouffe_ in his 1992 dissertation with offset 0
%t A006110 Table[QBinomial[n, 5, 2], {n, 5, 20}] (* _Vincenzo Librandi_, Aug 07 2016 *)
%o A006110 (Sage) [gaussian_binomial(n,5,2) for n in range(5,18)] # _Zerinvary Lajos_, May 24 2009
%o A006110 (Magma) r:=5; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // _Vincenzo Librandi_, Aug 07 2016
%Y A006110 Gaussian binomial coefficient [n, k] for q = 2: A000225 (k = 1), A006095 (k = 2), A006096 (k = 3), A006097 (k = 4), this sequence (k = 5), A022189 - A022195 ( k = 6 thru 12).
%K A006110 nonn,easy
%O A006110 5,2
%A A006110 _N. J. A. Sloane_