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%I #14 Sep 08 2022 08:46:15
%S 1,0,1,0,2,1,3,1,5,2,7,3,10,5,13,7,18,10,23,13,30,18,37,23,47,30,57,
%T 37,70,47,84,57,101,70,119,84,141,101,164,119,192,141,221,164,255,192,
%U 291,221,333,255,377,291,427,333,480,377,540,427,603,480,674,540,748,603,831,674,918,748,1014,831,1115,918,1226,1014,1342,1115
%N Molien series for invariants of finite Coxeter group D_5.
%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="/A266768/b266768.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1,1,0,-1,0,-1,-2,0,0,0,0,2,1,0,1,0,-1,-1,0,-1,0,1).
%F G.f.: 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)).
%F a(n) = a(n-2)+a(n-4)+a(n-5)-a(n-7)-a(n-9)-2*a(n-10)+2*a(n-15)+a(n-16)+a(n-18)-a(n-20)-a(n-21)-a(n-23)+a(n-25). - _Wesley Ivan Hurt_, May 03 2021
%p seq(coeff(series(1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)), x, n+1), x, n), n = 0..80); # _G. C. Greubel_, Jan 31 2020
%t CoefficientList[Series[1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)), {x,0,80}], x] (* _G. C. Greubel_, Jan 31 2020 *)
%o (PARI) my(x='x+O('x^80)); Vec(1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8))) \\ _G. C. Greubel_, Jan 31 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)) )); // _G. C. Greubel_, Jan 31 2020
%o (Sage)
%o def A266768_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)) ).list()
%o A266768_list(80) # _G. C. Greubel_, Jan 31 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 10 2016
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