A006100 Gaussian binomial coefficient [ n,2 ] for q=3. (Formerly M4912)

1, 13, 130, 1210, 11011, 99463, 896260, 8069620, 72636421, 653757313, 5883904390, 52955405230, 476599444231, 4289397389563, 38604583680520, 347441274648040, 3126971536402441

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

T. D. Noe, Table of n, a(n) for n=2..100

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 (13,-39,27).

Formula

G.f.: x^2/[(1-x)(1-3x)(1-9x)].

a(n) = (9^n - 4*3^n + 3)/48. - Mitch Harris, Mar 23 2008

a(n) = 4*a(n-1) -3*a(n-2) +9^(n-2), n>=4. - Vincenzo Librandi, Mar 20 2011

Maple

a:=n->sum((9^(n-j)-3^(n-j))/6, j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 15 2007
A006100:=-1/(z-1)/(3*z-1)/(9*z-1); # Simon Plouffe in his 1992 dissertation with offset 0

Mathematica

f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]    (* A203243 *)
Table[a[n]/3, {n, 2, 32}]  (* A006100 *)

Prog

(Sage) [gaussian_binomial(n, 2, 3) for n in range(2, 19)] # Zerinvary Lajos, May 25 2009

Crossrefs

Cf. A203243.

Sequence in context: A023061 A255495 A121033 * A037603 A037708 A142740

Adjacent sequences: A006097 A006098 A006099 * A006101 A006102 A006103

Keyword

nonn

Author

N. J. A. Sloane

Status

approved