A006101 Gaussian binomial coefficient [ n,3 ] for q=3.

1, 40, 1210, 33880, 925771, 25095280, 678468820, 18326727760, 494894285941, 13362799477720, 360801469802830, 9741692640081640, 263026177881648511, 7101711092201899360, 191746238094034963240, 5177148775980218655520, 139783020078437440101481

Offset: 3

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

• T. D. Noe, Table of n, a(n) for n=3..100
• 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 (40, -390, 1080, -729).

Programs

Magma

r:=3; q:=3; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 06 2016

Mathematica

Table[QBinomial[n, 3, 3], {n, 3, 20}] (* Vincenzo Librandi, Nov 06 2016 *)

Sage

[gaussian_binomial(n,3,3) for n in range(3,17)] # Zerinvary Lajos, May 25 2009

Formula

G.f.: z^3/((1-z)(1-3z)(1-9z)(1-27z)). Simon Plouffe in his 1992 dissertation

a(n) = (27^n - 13*9^n + 39*3^n - 27)/11232. - Mitch Harris, Mar 23 2008