%I A122370
%S A122370 1,5,29,172,1026,6134,36712,219847,1316963,7890594,47282065,283344410,
%T A122370 1698058817,10176618298,60990528210,365532989831,2190756912988,
%U A122370 13129979193808,78692862940748,471636719623539
%N A122370 Dimension of 6-variable non-commutative harmonics (twisted derivative). 
               The dimension of the space of non-commutative polynomials in 6 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 A122370 N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants 
               of the Symmetric Group in Noncommuting Variables, to appear Canad. 
               J. Math., arXiv:math.CO/0502082
%D A122370 C. Chevalley, Invariants of finite groups generated by reflections, Amer. 
               J. Math. 77 (1955), 778-782.
%D A122370 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%F A122370 o.g.f. (1-10*q+35*q^2-50*q^3+24*q^4)/(1-15*q+81*q^2-192*q^3+189*q^4-53*q^5) 
               more generally, sum( n!/(n-d)!*q^d/prod((1-r*q),r=1..d), d=0..n)/
               sum( q^d/prod((1-r*q),r=1..d), d=0..n) where n=6
%e A122370 a(1) = 5 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6 are all of degree 
               1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6
%p A122370 coeffs(convert(series((1-10*q+35*q^2-50*q^3+24*q^4)/(1-15*q+81*q^2-192*q^3+189*q^4-53*q^5),
               q,20),`+`)-O(q^20),q)
%Y A122370 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, 
               A122369, A122371, A122372.
%Y A122370 Sequence in context: A141812 A001653 A141814 this_sequence A088349 A137625 
               A083066
%Y A122370 Adjacent sequences: A122367 A122368 A122369 this_sequence A122371 A122372 
               A122373
%K A122370 nonn
%O A122370 0,2
%A A122370 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006