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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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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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)
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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.
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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
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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
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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 *)
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CROSSREFS
| Cf. A055105, A055106, A055107, A074664, A001519, A124293, A124294, A124295.
Sequence in context: A150188 A150189 A144169 * A129776 A129775 A054515
Adjacent sequences: A124289 A124290 A124291 * A124293 A124294 A124295
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KEYWORD
| nonn,changed
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AUTHOR
| Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 24 2006
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