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%I #22 Sep 08 2022 08:45:08
%S 1,1,3,5,10,14,23,31,46,59,80,100,130,157,196,233,283,330,392,451,527,
%T 599,689,776,883,985,1109,1229,1372,1510,1673,1831,2016,2195,2402,
%U 2604,2836,3061,3318,3569,3853,4130,4442,4747,5089,5423,5795,6160,6565,6961,7399
%N Expansion of Molien series for a certain 4-D group of order 48.
%C The first formula intersperses the terms with zeros, the second formula does not. - _Colin Barker_, Apr 02 2015
%H Colin Barker, <a href="/A078411/b078411.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_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-2,0,0,1,1,-1).
%F G.f.: (1 +x^4 +x^6 +2*x^8 +x^10 +x^12 +x^16)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)), even powers only.
%F G.f.: (1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/ ((1-x)^4*(1+x)^2*(1+x^2)*(1+x+x^2)). - _Colin Barker_, Apr 02 2015
%e G.f. = 1 + x^2 + 3*x^4 + 5*x^6 + 10*x^8 + 14*x^10 + 23*x^12 + 31*x^14 + 46*x^16 + ...
%p S:= series((1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/mul(1-x^j, j=1..4), x, 65):
%p seq(coeff(S, x, j), j = 0..60); # _G. C. Greubel_, Feb 02 2020
%t CoefficientList[Series[(1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), {x, 0, 60}], x] (* _G. C. Greubel_, Feb 02 2020 *)
%o (Magma) // Definition of group: F<al> := CyclotomicField(24); M := GeneralLinearGroup(4, F);
%o B1 := M![ -1, 0, 0, 0, 0,-1, 0, 0, 0, 0,-1, 0, 0, 0, 0,-1 ]; C1 := M![1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ];
%o C2 := M![0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1, 1, 0, 0, 0 ]; G := sub<M | B1, C1, C2 >;
%o (PARI) Vec((x^8+x^6+x^5+2*x^4+x^3+x^2+1) / ((x-1)^4*(x+1)^2*(x^2+1)*(x^2+x+1)) + O(x^60)) \\ _Colin Barker_, Apr 02 2015
%o (Sage)
%o def A078411_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) ).list()
%o A078411_list(60) # _G. C. Greubel_, Feb 02 2020
%Y Subgroup of the group in A078404.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 27 2002
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