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 A006122 Sum of Gaussian binomial coefficients [ n,k ] for q=8. (Formerly M2010) 5

%I M2010

%S 1,2,11,148,5917,617894,195118127,162366823096,409516908802369,

%T 2724882133766162378,54969878431787791720019,

%U 2925929849527072623051175132,472193512063977840212540697627493,201069312609841845828101079279279809006

%N Sum of Gaussian binomial coefficients [ n,k ] for q=8.

%C Generally, a(n) ~ c * q^(n^2/4), where c = EllipticTheta[3,0,1/q]/QPochhammer[1/q,1/q] if n is even and c = EllipticTheta[2,0,1/q]/QPochhammer[1/q,1/q] if n is odd. - _Vaclav Kotesovec_, Aug 21 2013

%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="/A006122/b006122.txt">Table of n, a(n) for n = 0..65</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)+(8^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - _Vaclav Kotesovec_, Aug 21 2013

%F a(n) ~ c * 8^(n^2/4), where c = EllipticTheta[3,0,1/8]/QPochhammer[1/8,1/8] = 1.455061175158... if n is even and c = EllipticTheta[2,0,1/8]/QPochhammer[1/8,1/8] = 1.405381182498... if n is odd. - _Vaclav Kotesovec_, Aug 21 2013

%t Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(8^(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, 8], {k, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 13 2016 *)

%o (MAGMA) [n le 2 select n else 2*Self(n-1)+(8^(n-2)-1)*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Aug 13 2016

%K easy,nonn

%O 0,2

%A _N. J. A. Sloane_

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Last modified October 19 11:26 EDT 2019. Contains 328216 sequences. (Running on oeis4.)