A006105 Gaussian binomial coefficient [ n,2 ] for q=4. (Formerly M5115)

1, 21, 357, 5797, 93093, 1490853, 23859109, 381767589, 6108368805, 97734250405, 1563749404581, 25019996065701, 400319959420837, 6405119440211877, 102481911401303973

Offset

2,2

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 = 2..200

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 (21,-84,64)

Formula

G.f.: x^2/((1-x)*(1-4*x)*(1-16*x)). [Multiplied by x^2 to match offset by R. J. Mathar, Mar 11 2009]

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

Maple

A006105:=-1/(z-1)/(4*z-1)/(16*z-1); # Simon Plouffe in his 1992 dissertation, assuming offset zero

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}] (* 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 *)
LinearRecurrence[{21, -84, 64}, {1, 21, 357}, 20] (* Harvey P. Dale, Feb 17 2020 *)

Prog

(Sage) [gaussian_binomial(n, 2, 4) for n in range(2, 17)] # Zerinvary Lajos, May 28 2009

Crossrefs

Sequence in context: A271633 A184289 A192093 * A167032 A051564 A302308

Adjacent sequences: A006102 A006103 A006104 * A006106 A006107 A006108

Keyword

nonn

Author

N. J. A. Sloane

Status

approved