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A006098 Gaussian binomial coefficient [ 2n,n ] for q=2.
(Formerly M3138)
3

%I M3138

%S 1,3,35,1395,200787,109221651,230674393235,1919209135381395,

%T 63379954960524853651,8339787869494479328087443,

%U 4380990637147598617372537398675,9196575543360038413217351554014467475,77184136346814161837268404381760884963259795

%N Gaussian binomial coefficient [ 2n,n ] 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="/A006098/b006098.txt">Table of n, a(n) for n = 0..35</a>

%H I. Siap and I. Aydogdu, <a href="http://arxiv.org/abs/1303.6985">Counting The Generator Matrices of Z_2 Z_8 Codes</a>, arXiv:1303.6985 [math.CO], 2013.

%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>.

%F a(n) = A022166(2n,n). - _Alois P. Heinz_, Mar 30 2016

%F a(n) ~ c * 2^(n^2), where c = A065446. - _Vaclav Kotesovec_, Sep 22 2016

%F a(n) = Sum_{k=0..n} 2^(k^2)*(A022166(n,k))^2. - _Werner Schulte_, Mar 09 2019

%t Table[QBinomial[2n,n,2],{n,0,20}] (* _Harvey P. Dale_, Oct 22 2012 *)

%o (Sage) [gaussian_binomial(2*n,n,2) for n in xrange(0,11)] # _Zerinvary Lajos_, May 25 2009

%o (PARI) q=2; {a(n) = prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1))) };

%o vector(10, n, n--; a(n)) \\ _G. C. Greubel_, Mar 09 2019

%o (MAGMA) q:=2; [n le 0 select 1 else (&*[(1-q^(2*n-j))/(1-q^(j+1)): j in [0..n-1]]): n in [0..15]]; // _G. C. Greubel_, Mar 09 2019

%Y Cf. A022166.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Harvey P. Dale_, Oct 22 2012

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Last modified April 24 19:49 EDT 2019. Contains 322446 sequences. (Running on oeis4.)