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Molien series for invariants of finite Coxeter group D_11.
1

%I #9 Sep 08 2022 08:46:15

%S 1,0,1,0,2,0,3,0,5,0,7,1,11,1,15,2,22,3,30,5,42,7,56,11,76,15,99,22,

%T 131,30,169,42,219,56,278,76,355,99,445,131,560,169,695,219,863,278,

%U 1060,355,1303,445,1586,560,1930,695,2331,863,2812,1060,3370,1303,4035,1586,4802,1930,5708,2331,6751,2812,7972,3370,9373,4035,11004

%N Molien series for invariants of finite Coxeter group D_11.

%C The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.

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

%H G. C. Greubel, <a href="/A266774/b266774.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^10)*(1-t^11)*(1-t^12)*(1-t^14)*(1-t^16)*(1-t^18)*(1-t^20)).

%p seq(coeff(series(1/((1-x^11)*mul(1-x^(2*j), j=1..10)), x, n+1), x, n), n = 0..80); # _G. C. Greubel_, Feb 03 2020

%t CoefficientList[Series[1/((1-x^11)*Product[1-x^(2*j), {j,10}]), {x,0,80}], x] (* _G. C. Greubel_, Feb 03 2020 *)

%o (PARI) Vec(1/((1-x^11)*prod(j=1,10,1-x^(2*j))) +O('x^80)) \\ _G. C. Greubel_, Feb 03 2020

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^11)*(&*[1-x^(2*j): j in [1..10]])) )); // _G. C. Greubel_, Feb 03 2020

%o (Sage)

%o def A266774_list(prec):

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

%o return P( 1/((1-x^11)*product(1-x^(2*j) for j in (1..10))) ).list()

%o A266774_list(80) # _G. C. Greubel_, Feb 03 2020

%Y Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.

%K nonn

%O 0,5

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