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A006105
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Gaussian binomial coefficient [ n,2 ] for q=4.
(Formerly M5115)
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9
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1, 21, 357, 5797, 93093, 1490853, 23859109, 381767589, 6108368805, 97734250405, 1563749404581, 25019996065701, 400319959420837, 6405119440211877, 102481911401303973
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OFFSET
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2,2
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REFERENCES
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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.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 2..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.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64)
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FORMULA
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G.f.: x^2/((1-x)*(1-4*x)*(1-16*x)).
a(n) = (16^n - 5*4^n + 4)/180 - Mitch Harris, Mar 23 2008
a(n) = 5*a(n-1) -4*a(n-2) +16^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011
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MAPLE
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A006105:=-1/(z-1)/(4*z-1)/(16*z-1); [Simon Plouffe in his 1992 dissertation, assuming offset zero.]
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MATHEMATICA
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faq[n_, q_] = Product[(1-q^(1+k))/(1-q), {k, 0, n-1}];
qbin[n_, m_, q_] = faq[n, q]/(faq[m, q]*faq[n-m, q]);
Table[qbin[n, 2, 4], {n, 2, 16}] (* Jean-François Alcover, Jul 21 2011 *)
CoefficientList[Series[1 / ((1 - x) (1 - 4 x) (1 - 16 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2013 *)
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PROG
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(Sage) [gaussian_binomial(n, 2, 4) for n in xrange(2, 17)] # [From Zerinvary Lajos, May 28 2009]
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CROSSREFS
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Sequence in context: A201878 A184289 A192093 * A167032 A051564 A108495
Adjacent sequences: A006102 A006103 A006104 * A006106 A006107 A006108
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Multiplied g.f. by x^2 to match offset R. J. Mathar, Mar 11 2009
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STATUS
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approved
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