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A006106 Gaussian binomial coefficient [ n,3 ] for q = 4.
(Formerly M5360)
2
1, 85, 5797, 376805, 24208613, 1550842085, 99277752549, 6354157930725, 406672215935205, 26027119554103525, 1665737215212030181, 106607206793565997285, 6822861635108183247077, 436663151052043168024805, 27946441769812674154891493 (list; graph; refs; listen; history; text; internal format)
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

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

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

MATHEMATICA

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

PROG

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

(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

CROSSREFS

Sequence in context: A201796 A093285 A011813 * A015338 A181015 A131750

Adjacent sequences:  A006103 A006104 A006105 * A006107 A006108 A006109

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 23 12:43 EDT 2019. Contains 321430 sequences. (Running on oeis4.)