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%I M1812 #35 Sep 08 2022 08:44:34
%S 1,2,7,44,529,12278,565723,51409856,9371059621,3387887032202,
%T 2463333456292207,3557380311703796564,10339081666350180289849,
%U 59703612489554311631068958
%N Sum of Gaussian binomial coefficients [ n,k ] for q=4.
%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 Vincenzo Librandi, <a href="/A006118/b006118.txt">Table of n, a(n) for n = 0..80</a>
%H S. Hitzemann, W. Hochstattler, <a href="http://dx.doi.org/10.1016/j.disc.2010.09.001">On the combinatorics of Galois numbers</a>, Discr. Math. 310 (2010) 3551-3557.
%H Kent E. Morrison, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H M. Sved, <a href="/A006095/a006095.pdf">Gaussians and binomials</a>, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
%F a(n) = 2*a(n-1)+(4^(n-1)-1)*a(n-2), n>1. [Hitzemann and Hochstattler]. - _R. J. Mathar_, Aug 21 2013
%F a(n) ~ c * 4^(n^2/4), where c = EllipticTheta[3,0,1/4]/QPochhammer[1/4,1/4] = 2.189888057761... if n is even and c = EllipticTheta[2,0,1/4]/QPochhammer[1/4,1/4] = 2.182810929357... if n is odd. - _Vaclav Kotesovec_, Aug 21 2013
%t Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(4^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* _Vaclav Kotesovec_, Aug 21 2013 *)
%t Table[Sum[QBinomial[n, k, 4], {k, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 13 2016 *)
%o (Magma) [n le 2 select n else 2*Self(n-1)+(4^(n-2)-1)*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Aug 13 2016
%Y Row sums of triangle A022168.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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