A006100 Gaussian binomial coefficient [n, 2] for q = 3.

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

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).

Crossrefs

Cf. A203243.

Programs

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 *)

Sage

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

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

From Peter Bala, Jul 01 2025: (Start)

G.f. with an offset of 0: exp(Sum_{n >= 1} b(3*n)/b(n)*x^n/n) = 1 + 13*x + 130*x^2 + ..., where b(n) = 3^n - 1.

The following series telescope:

Sum_{n >= 2} 3^n/a(n) = 12; Sum_{n >= 2} 3^n/(a(n)*a(n+3)) = 129/16900;

Sum_{n >= 2} 9^n/(a(n)*a(n+3)) = 1227/16900;

Sum_{n >= 2} 3^n/(a(n)*a(n+3)*a(n+6)) = 156706257/18829431219368770. (End)