%I A122369
%S A122369 1,4,19,93,459,2273,11274,55964,277924,1380527,6858356,34074280,
%T A122369 169297743,841173845,4179517118,20766807551,103184684826,512698227699,
%U A122369 2547469553647,12657750705603,62893284231103,312501512711984
%N A122369 Dimension of 5-variable non-commutative harmonics (twisted derivative). 
               The dimension of the space of non-commutative polynomials in 5 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 A122369 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 A122369 C. Chevalley, Invariants of finite groups generated by reflections, Amer. 
               J. Math. 77 (1955), 778-782.
%D A122369 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%F A122369 o.g.f. (1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4) 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=5
%e A122369 a(1) = 4 because x1-x2, x2-x3, x3-x4, x4-x5 are all of degree 1 and are 
               killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5
%p A122369 coeffs(convert(series((1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4),
               q,30),`+`)-O(q^30),q)
%Y A122369 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, 
               A122370, A122371, A122372.
%Y A122369 Sequence in context: A151253 A121179 A131552 this_sequence A005978 A083065 
               A137636
%Y A122369 Adjacent sequences: A122366 A122367 A122368 this_sequence A122370 A122371 
               A122372
%K A122369 nonn
%O A122369 0,2
%A A122369 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006