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Molien series for invariants of finite Coxeter group A_7.
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%I #15 Oct 21 2022 22:06:55

%S 1,0,1,1,2,2,4,4,7,7,11,12,18,19,27,30,40,44,58,64,82,91,113,126,155,

%T 171,207,230,274,303,358,395,462,509,589,649,746,818,934,1024,1161,

%U 1269,1432,1562,1753,1909,2131,2317,2577,2794,3095,3352,3698,3997,4396,4743,5200,5601,6121,6584,7177,7705,8377,8983,9741,10429,11285,12065

%N Molien series for invariants of finite Coxeter group A_7.

%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="/A266776/b266776.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_35">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 1, 0, 0, -1, -1, -2, -2, -1, 1, 2, 2, 3, 2, 1, -1, -2, -3, -2, -2, -1, 1, 2, 2, 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)).

%t CoefficientList[Series[1/Product[1-t^k, {k,2,8}], {t, 0, 40}], t] (* _G. C. Greubel_, Oct 24 2018 *)

%o (PARI) t='t+O('t^40); Vec(1/prod(k=2,8, 1-t^k)) \\ _G. C. Greubel_, Oct 24 2018

%o (Magma) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[1-t^k: k in [2..8]]))); // _G. C. Greubel_, Oct 24 2018

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

%K nonn,easy

%O 0,5

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