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A006121 Sum of Gaussian binomial coefficients [ n,k ] for q=7.
(Formerly M1984)
5

%I M1984

%S 1,2,10,116,3652,285704,61946920,33736398032,51083363186704,

%T 194585754101247008,2061787082699360148640,54969782721182164414355264

%N Sum of Gaussian binomial coefficients [ n,k ] for q=7.

%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="/A006121/b006121.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)+(7^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - _Vaclav Kotesovec_, Aug 21 2013

%F a(n) ~ c * 7^(n^2/4), where c = EllipticTheta[3,0,1/7]/QPochhammer[1/7,1/7] = 1.537469386940... if n is even and c = EllipticTheta[2,0,1/7]/QPochhammer[1/7,1/7] = 1.499386995418... if n is odd. - _Vaclav Kotesovec_, Aug 21 2013

%t Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(7^(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, 7], {k, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 13 2016 *)

%o (MAGMA) [n le 2 select n else 2*Self(n-1)+(7^(n-2)-1)*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 13 2016

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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Last modified April 20 17:54 EDT 2019. Contains 322310 sequences. (Running on oeis4.)