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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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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.
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (21,-84,64)
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FORMULA
| 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
| A006105:=-1/(z-1)/(4*z-1)/(16*z-1); [S. Plouffe in his 1992 dissertation, assuming offset zero.]
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MATHEMATICA
| 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}] (* From Jean-François Alcover, Jul 21 2011 *)
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PROG
| (Sage) [gaussian_binomial(n, 2, 4) for n in xrange(2, 17)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
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CROSSREFS
| Sequence in context: A201878 A184289 A192093 * A167032 A051564 A108495
Adjacent sequences: A006102 A006103 A006104 * A006106 A006107 A006108
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Multiplied g.f. by x^2 to match offset R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
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