A006096 Gaussian binomial coefficient [ n,3 ] for q=2. (Formerly M4982)

1, 15, 155, 1395, 11811, 97155, 788035, 6347715, 50955971, 408345795, 3269560515, 26167664835, 209386049731, 1675267338435, 13402854502595, 107225699266755, 857817047249091, 6862582190715075, 54900840777134275, 439207459223777475

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

T. D. Noe, Table of n, a(n) for n=3..203

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 (15,-70,120,-64).

Formula

G.f.: x^3/((1-x)(1-2x)(1-4x)(1-8x)).

(With a different offset) a(n)=(-1+7*2^n-14*4^n+8*8^n)/21. - James R. Buddenhagen, Dec 14 2003

Maple

seq((-1+7*2^n-14*4^n+8*8^n)/21, n=1..20);
A006096:=1/(z-1)/(8*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Mathematica

Drop[CoefficientList[Series[x^3/((1 - x) (1 - 2 x) (1 - 4 x) (1 - 8 x)), {x, 0, 30}], x], 3]
QBinomial[Range[3, 30], 3, 2] (* Harvey P. Dale, Jan 28 2013 *)

Prog

(Sage) [gaussian_binomial(n, 3, 2) for n in range(3, 23)] # Zerinvary Lajos, May 24 2009
(Magma) r:=3; q:=2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 06 2016

Crossrefs

Sequence in context: A098685 A223995 A323971 * A346843 A341918 A099915

Adjacent sequences: A006093 A006094 A006095 * A006097 A006098 A006099

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved