|
%I #9 Sep 08 2022 08:46:15
%S 1,0,1,0,2,0,3,0,5,1,7,1,11,2,15,3,22,5,30,7,41,11,54,15,73,22,94,30,
%T 123,41,157,54,201,73,252,94,318,123,393,157,488,201,598,252,732,318,
%U 887,393,1076,488,1291,598,1549,732,1845,887,2194,1076,2592,1291,3060,1549,3589,1845,4206,2194,4904,2592,5708,3060,6615,3589
%N Molien series for invariants of finite Coxeter group D_9.
%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="/A266772/b266772.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^9)*(1-t^10)*(1-t^12)*(1-t^14)*(1-t^16)).
%p seq(coeff(series(1/((1-x^9)*mul(1-x^(2*j), j=1..8)), x, n+1), x, n), n = 0..80); # _G. C. Greubel_, Feb 03 2020
%t CoefficientList[Series[1/((1-x^9)*Product[1-x^(2*j), {j,8}]), {x,0,80}], x] (* _G. C. Greubel_, Feb 03 2020 *)
%o (PARI) Vec(1/((1-x^9)*prod(j=1,8,1-x^(2*j))) +O('x^80)) \\ _G. C. Greubel_, Feb 03 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^9)*(&*[1-x^(2*j): j in [1..8]])) )); // _G. C. Greubel_, Feb 03 2020
%o (Sage)
%o def A266772_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/((1-x^9)*product(1-x^(2*j) for j in (1..8))) ).list()
%o A266772_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
|
