%I
%S 1,6,41,285,1989,13901,97215,680079,4758408,33297267,233014444,
%T 1630701426,11412409945,79870754268,558989013403,3912210491549,
%U 27380636068267,191631324294463,1341190961828143,9386756237545989
%N Dimension of 7variable noncommutative harmonics (twisted derivative). The dimension of the space of noncommutative polynomials in 7 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).
%D C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778782.
%D M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626637.
%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.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266296.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,170,669,1314,1157,309).
%F G.f.: (115*q+ 85*q^2225*q^3+274*q^4120*q^5) / (121*q+170*q^2669*q^3 +1314*q^41157*q^5 +309*q^6) more generally, sum( n!/(nd)!*q^d/prod((1r*q),r=1..d), d=0..n)/sum( q^d/prod((1r*q), r=1..d), d=0..n) where n=7.
%e a(1) = 6 because x1x2, x2x3, x3x4, x4x5, x5x6, x6x7 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6+d_x7.
%p coeffs(convert(series((115*q+ 85*q^2225*q^3+274*q^4120*q^5) / (121*q+170*q^2669*q^3+1314*q^41157*q^5+309*q^6),q,20),`+`)O(q^20),q);
%t LinearRecurrence[{21, 170, 669, 1314, 1157, 309}, {1, 6, 41, 285, 1989, 13901}, 20] (* _JeanFrançois Alcover_, Sep 22 2017 *)
%Y Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122372.
%K nonn
%O 0,2
%A _Mike Zabrocki_, Aug 30 2006
