A124292 revision #19
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A124292
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Number of free generators of degree n of symmetric polynomials in 4 noncommuting variables.
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11
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1, 1, 2, 6, 21, 78, 297, 1143, 4419, 17118, 66366, 257391, 998406, 3873015, 15024609, 58285737, 226111986, 877174110, 3402893997, 13201132950, 51212274057, 198672129783, 770725711035, 2989941920334, 11599136512038, 44997518922327, 174562710686622
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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Stanley, Richard P., Enumerative combinatorics. Vol. 1.Cambridge University Press, Cambridge, 1997. pages 96-100
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LINKS
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FORMULA
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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
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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 24 2006
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STATUS
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editing
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