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%I M1952 #27 Sep 08 2022 08:44:34
%S 1,2,9,88,2111,118182,16649389,5547079988,4671840869691,
%T 9326302435784002,47100039978152210249,564020035264998031552848,
%U 17088883834526416216141122391,1227783027118593811726444427584862,223195138386683651821176756496371359589
%N Sum of Gaussian binomial coefficients [ n,k ] for q=6.
%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="/A006120/b006120.txt">Table of n, a(n) for n = 0..70</a>
%H Kent E. Morrison, <a href="http://www.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)+(6^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - _Vaclav Kotesovec_, Aug 21 2013
%F a(n) ~ c * 6^(n^2/4), where c = EllipticTheta[3,0,1/6]/QPochhammer[1/6,1/6] = 1.656816524577... if n is even and c = EllipticTheta[2,0,1/6]/QPochhammer[1/6,1/6] = 1.630173070572... if n is odd. - _Vaclav Kotesovec_, Aug 21 2013
%t Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(6^(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, 6], {k, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 13 2016 *)
%o (Magma) [n le 2 select n else 2*Self(n-1)+(6^(n-2)-1)*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 13 2016
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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