%I A122368
%S A122368 1,3,11,42,162,627,2430,9423,36549,141777,549990,2133594,8276985,
%T A122368 32109534,124565121,483235875,1874657763,7272519066,28212902154,
%U A122368 109448714619,424593725526,1647162628047,6389978382405,24789187818585
%N A122368 Dimension of 4-variable non-commutative harmonics (twisted derivative). 
               The dimension of the space of non-commutative polynomials in 4 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 A122368 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 A122368 C. Chevalley, Invariants of finite groups generated by reflections, Amer. 
               J. Math. 77 (1955), 778-782.
%D A122368 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%F A122368 o.g.f. (1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3) 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=4
%e A122368 a(1) = 3 because x1-x2, x2-x3, x3-x4 are all of degree 1 and are killed 
               by the differential operator d_x1+d_x2+d_x3+d_x4
%p A122368 coeffs(convert(series((1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3),q,30),`+`)-O(q^30),
               q);
%Y A122368 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, 
               A122370, A122371, A122372.
%Y A122368 Sequence in context: A077830 A106460 A059716 this_sequence A032443 A143464 
               A117641
%Y A122368 Adjacent sequences: A122365 A122366 A122367 this_sequence A122369 A122370 
               A122371
%K A122368 nonn
%O A122368 1,2
%A A122368 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006