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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
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)
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 range(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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified April 23 15:20 EDT 2024. Contains 371916 sequences. (Running on oeis4.)