%I #25 Mar 03 2024 14:29:47
%S 1,1,2,6,22,92,426,2145,11589,66425,399682,2500037,16115347,106266473,
%T 712602272,4837372576,33128183406,228308233098,1580495251012,
%U 10976092266889,76398165848091,532614662149795,3717370694711130
%N Number of free generators of degree n of symmetric polynomials in 7-noncommuting variables.
%C Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=7
%H N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="https://arxiv.org/abs/math/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
%H M. C. Wolf, <a href="https://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_06">Index entries for linear recurrences with constant coefficients</a>, signature (21, -170, 669, -1314, 1157, -309).
%F O.g.f.: (1-20*q+151*q^2-535*q^3+881*q^4-531*q^5) / (1-21*q+170*q^2 -669*q^3 +1314*q^4-1157*q^5+309*q^6) = (1 - 1/(Sum_{k=0..7} q^k/(prod_{i=1}^k (1-i*q))))/q.
%F a(n) = add( A055105(n,k), k=1..7) = add(A055106(n,k), k=1..6).
%t LinearRecurrence[{21, -170, 669, -1314, 1157, -309}, {1, 1, 2, 6, 22, 92}, 23] (* _Jean-François Alcover_, Jan 27 2019 *)
%Y Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124293, A124294.
%K nonn
%O 1,3
%A _Mike Zabrocki_, Oct 24 2006