%I A122372
%S A122372 1,7,55,438,3498,27962,223604,1788406,14305102,114429193,915366442,
%T A122372 7322521512,58577537621,468602617723,3748697751384,29988696932490,
%U A122372 239903055854075,1919175464438065,15353030007717639,122821355074655309
%N A122372 Dimension of 8-variable non-commutative harmonics (twisted derivative). 
               The dimension of the space of non-commutative polynomials in 8 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).
%C A122372 coeffs(convert(series((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/
               (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7),q,
               20),`+`)-O(q^20),q)
%D A122372 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 A122372 C. Chevalley, Invariants of finite groups generated by reflections, Amer. 
               J. Math. 77 (1955), 778-782.
%D A122372 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%F A122372 o.g.f. (1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/(1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^\
               6-2119*q^7) 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=8
%e A122372 A122371 a(1) = 7 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7, x7-x8 
               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 A122372 coeffs(convert(series((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/
               (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7),q,
               20),`+`)-O(q^20),q)
%Y A122372 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, 
               A122369, A122370, A122371.
%Y A122372 Sequence in context: A069404 A015564 A070997 this_sequence A083068 A097189 
               A049028
%Y A122372 Adjacent sequences: A122369 A122370 A122371 this_sequence A122373 A122374 
               A122375
%K A122372 nonn
%O A122372 0,2
%A A122372 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006