A006121 Sum of Gaussian binomial coefficients [ n,k ] for q=7.

1, 2, 10, 116, 3652, 285704, 61946920, 33736398032, 51083363186704, 194585754101247008, 2061787082699360148640, 54969782721182164414355264

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• 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.
• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..65
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Programs

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

Mathematica

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 *)
Table[Sum[QBinomial[n, k, 7], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

Formula

a(n) = 2*a(n-1)+(7^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

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