cp's OEIS Frontend

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).

%I A122370 #15 Sep 22 2017 11:38:12
%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 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.
%H A122370 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
%H A122370 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-81,192,-189,53)
%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)
%t A122370 LinearRecurrence[{15, -81, 192, -189, 53}, {1, 5, 29, 172, 1026}, 20] (* _Jean-François Alcover_, Sep 22 2017 *)
%Y A122370 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122371, A122372.
%K A122370 nonn
%O A122370 0,2
%A A122370 _Mike Zabrocki_, Aug 30 2006