%I #13 Nov 17 2018 15:11:35
%S 1,4,19,93,459,2273,11274,55964,277924,1380527,6858356,34074280,
%T 169297743,841173845,4179517118,20766807551,103184684826,512698227699,
%U 2547469553647,12657750705603,62893284231103,312501512711984,1552744642741738,7715214279423070
%N 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 C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
%D M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
%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, 266-296
%F 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 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 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);
%t gf = With[{n = 5}, Sum[n!/(n-d)! q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}]/Sum[ q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}]]; CoefficientList[gf + O[q]^22, q] (* _Jean-François Alcover_, Nov 17 2018 *)
%Y Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.
%K nonn
%O 0,2
%A _Mike Zabrocki_, Aug 30 2006