OFFSET
1,3
COMMENTS
Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=4.
Also the number of nonisomorphic graded posets with 0 and 1 of rank n with no 3-element antichain. (Richard Stanley, Nov 30 2011)
Also the number of nonisomorphic graded posets with 0 of rank n+1 with no 3-element antichain. (Using Stanley's definition of graded, that all maximal chains have length n.) -David Nacin, Feb 26 2012
REFERENCES
Stanley, Richard P., Enumerative combinatorics. Vol. 1.Cambridge University Press, Cambridge, 1997. pages 96-100
LINKS
N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables , MR2398749, Cand. J. Math 60 (2008) 266-296.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
FORMULA
O.g.f.: (1-5q+5q^2)/(1-6q+9q^2-3q^3) = 1 - 1/(sum_{k=0}^4 q^k/(prod_{i=1}^k (1-i*q))).
a(n) = 6a(n-1) - 9a(n-2) + 3a(n-3). - David Nacin (nacind(AT)wpunj.edu), Feb 11 2012
a(n) = A055105(n,1) + A055105(n,2) + A055105(n,3) + A055105(n,4) = A055106(n,1) + A055106(n,2) + A055106(n,3).
Given matrix A = [[2,1,1],[1,3,0],[1,1,1]], a(n+1) = top left entry in A^n. - David Nacin, Feb 11 2012
MAPLE
a:= n-> (Matrix([[2, 1, 1]]). Matrix(3, (i, j)-> if i=j-1 then 1 elif j=1 then [6, -9, 3][i] else 0 fi)^(n-1))[1, 3]: seq (a(n), n=1..26); # Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 05 2008
MATHEMATICA
m = {{2, 1, 1}, {1, 3, 0}, {1, 1, 1}}; Table[MatrixPower[m, n][[1, 1]], {n, 0, 40}] (* David Nacin, Feb 11 2012 *)
LinearRecurrence[{6, -9, 3}, {1, 1, 2}, 70] (* From Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 24 2006
STATUS
proposed