A006101 Gaussian binomial coefficient [ n,3 ] for q=3. (Formerly M5272)

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

Offset

3,2

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).

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

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A229457 A331350 A061650 * A278381 A251202 A239189

Adjacent sequences: A006098 A006099 A006100 * A006102 A006103 A006104

Keyword

nonn

Author

N. J. A. Sloane

Status

approved