A266777 - OEIS

%I #12 Feb 23 2022 23:06:04

%S 1,0,1,1,2,2,4,4,7,8,11,13,19,21,29,34,44,51,66,75,95,110,134,155,189,

%T 215,258,296,349,398,468,529,617,698,804,907,1042,1167,1332,1492,1690,

%U 1886,2130,2366,2660,2951,3298,3649,4069,4484,4981,5482,6064,6657,7347,8041,8849,9670,10605,11565,12659,13769,15034,16330,17782,19278,20955

%N Molien series for invariants of finite Coxeter group A_8.

%C The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i).

%C Note that this is the root system A_k, not the alternating group Alt_k.

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

%H Ray Chandler, <a href="/A266777/b266777.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_44">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,0,0,-1,-1,-1,-2,-2,0,1,2,3,3,2,1,0,-2,-3,-4,-3,-2,0,1,2,3,3,2,1,0,-2,-2,-1,-1,-1,0,0,1,1,1,0,-1).

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%F G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)).

%p seq(coeff(series( mul(1/(1-x^j), j=2..9), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Feb 01 2020

%t CoefficientList[Series[Product[1/(1-x^j), {j,2,9}], {x,0,70}], x] (* _G. C. Greubel_, Feb 01 2020 *)

%o (PARI) Vec( prod(j=2,9, 1/(1-x^j)) + O('x^70) ) \\ _G. C. Greubel_, Feb 01 2020

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (&*[1/(1-x^j): j in [2..9]]) )); // _G. C. Greubel_, Feb 01 2020

%o (Sage)

%o def A266777_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( product(1/(1-x^j) for j in (2..9)) ).list()

%o A266777_list(70) # _G. C. Greubel_, Feb 01 2020

%Y Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Jan 11 2016