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
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 13.
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved