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%I #28 Sep 08 2022 08:45:28
%S 1,1,2,6,22,91,406,1896,9093,44279,217500,1073657,5314870,26352107,
%T 130778039,649352929,3225196431,16021584848,79597062632,395469296912,
%U 1964908443531,9762920818182,48508934285620,241027326818991,1197601448443963,5950578465799856
%N Number of free generators of degree n of symmetric polynomials in 5-noncommuting variables.
%C Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=5
%H Alois P. Heinz, <a href="/A124293/b124293.txt">Table of n, a(n) for n = 1..500</a>
%H N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="http://arxiv.org/abs/math.CO/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math/0502082 [math.CO], 2005.
%H N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="http://dx.doi.org/10.4153/CJM-2008-013-4">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, Canad. J. Math. 60 (2008), no. 2, 266-296.
%H M. C. Wolf, <a href="http://dx.doi.org/10.1215/S0012-7094-36-00253-3">Symmetric functions of noncommutative elements</a>, Duke Math. J. 2 (1936), 626-637.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-32,37,-11).
%F O.g.f. (1-9q+24q^2-19q^3)/(1-10q+32q^2-37q^3+11q^4) = (1 - 1/(sum_{k=0}^5 q^k/(prod_{i=1}^k (1-i*q))))/q a(n) = add( A055105(n,k), k=1..5) = add(A055106(n,k),k=1..4)
%p a:= n-> (Matrix([[6,2,1,1]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [10, -32, 37, -11][i] else 0 fi)^(n-1))[1,4]: seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 05 2008
%t LinearRecurrence[{10, -32, 37, -11}, {1, 1, 2, 6}, 30] (* _Jean-François Alcover_, Jan 08 2016 *)
%o (Magma) I:=[1,1,2,6]; [n le 4 select I[n] else 10*Self(n-1)-32*Self(n-2)+37*Self(n-3)-11*Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Jan 09 2016
%Y Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124294, A124295.
%K nonn
%O 1,3
%A _Mike Zabrocki_, Oct 24 2006
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