|
| |
|
|
A006100
|
|
Gaussian binomial coefficient [ n,2 ] for q=3.
(Formerly M4912)
|
|
5
| |
|
|
1, 13, 130, 1210, 11011, 99463, 896260, 8069620, 72636421, 653757313, 5883904390, 52955405230, 476599444231, 4289397389563, 38604583680520, 347441274648040, 3126971536402441
(list; graph; refs; listen; history; internal format)
|
|
|
|
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
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with 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 (maharri(AT)gmail.com), 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 (zerinvarylajos(AT)yahoo.com), Jan 15 2007
A006100:=-1/(z-1)/(3*z-1)/(9*z-1); [S. 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
| (Other) sage: [gaussian_binomial(n, 2, 3) for n in xrange(2, 19)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 25 2009]
|
|
|
CROSSREFS
| Cf. A203243.
Sequence in context: A155623 A023061 A121033 * A037603 A037708 A142740
Adjacent sequences: A006097 A006098 A006099 * A006101 A006102 A006103
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
