A006122 Sum of Gaussian binomial coefficients [ n,k ] for q=8.

1, 2, 11, 148, 5917, 617894, 195118127, 162366823096, 409516908802369, 2724882133766162378, 54969878431787791720019, 2925929849527072623051175132, 472193512063977840212540697627493, 201069312609841845828101079279279809006

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

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

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)+(8^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016

Mathematica

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 *)
Table[Sum[QBinomial[n, k, 8], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

Formula

a(n) = 2*a(n-1)+(8^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

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