A006106 Gaussian binomial coefficient [ n,3 ] for q = 4.

1, 85, 5797, 376805, 24208613, 1550842085, 99277752549, 6354157930725, 406672215935205, 26027119554103525, 1665737215212030181, 106607206793565997285, 6822861635108183247077, 436663151052043168024805, 27946441769812674154891493

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 = 3..200
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
• Index entries for linear recurrences with constant coefficients, signature (85, -1428, 5440, -4096).

Programs

Magma

r:=3; q:=4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Aug 07 2016

Mathematica

Table[QBinomial[n, 3, 4], {n, 3, 20}] (* Vincenzo Librandi, Aug 07 2016 *)

Sage

[gaussian_binomial(n,3,4) for n in range(3,15)] # Zerinvary Lajos, May 27 2009

Formula

G.f.: x^3/((1-x)*(1-4*x)*(1-16*x)*(1-64*x)). - Simon Plouffe in his 1992 dissertation

a(n) = Product_{i=1..3} (4^(n-i+1)-1)/(4^i-1), by definition. - Vincenzo Librandi, Aug 07 2016