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Molien series for 3-dimensional group [2+,n ] = 2*(n/2).
1

%I #19 Sep 08 2022 08:44:36

%S 1,0,2,0,3,0,4,1,5,2,6,3,8,4,10,5,12,6,14,8,16,10,18,12,21,14,24,16,

%T 27,18,30,21,33,24,36,27,40,30,44,33,48,36,52,40,56,44,60,48,65,52,70,

%U 56,75,60,80,65,85,70,90

%N Molien series for 3-dimensional group [2+,n ] = 2*(n/2).

%H G. C. Greubel, <a href="/A008735/b008735.txt">Table of n, a(n) for n = 0..10000</a>

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

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1).

%F G.f.: (1+x^7)/((1-x^2)^2*(1-x^12)).

%F a(n)= +a(n-1) +a(n-2) -a(n-3) +a(n-12) -a(n-13) -a(n-14) +a(n-15). - _R. J. Mathar_, Dec 18 2014

%p seq(coeff(series((1+x^7)/((1-x^2)^2*(1-x^12)), x, n+1), x, n), n = 0 .. 70); # modified by _G. C. Greubel_, Jul 30 2019

%t CoefficientList[Series[(1+x^7)/((1-x^2)^2*(1-x^12)), {x,0,70}], x] (* _G. C. Greubel_, Jul 30 2019 *)

%t LinearRecurrence[{1,1,-1,0,0,0,0,0,0,0,0,1,-1,-1,1},{1,0,2,0,3,0,4,1,5,2,6,3,8,4,10},60] (* _Harvey P. Dale_, Aug 10 2019 *)

%o (PARI) my(x='x+O('x^70)); Vec((1+x^7)/((1-x^2)^2*(1-x^12))) \\ _G. C. Greubel_, Jul 30 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^7)/((1-x^2)^2*(1-x^12)) )); // _G. C. Greubel_, Jul 30 2019

%o (Sage) ((1+x^7)/((1-x^2)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 30 2019

%o (GAP) a:=[1,0,2,0,3,0,4,1,5,2,6,3,8,4,10];; for n in [16..70] do a[n]:=a[n-1]+a[n-2]-a[n-3]+a[n-12]-a[n-13]-a[n-14]+a[n-15]; od; a; # _G. C. Greubel_, Jul 30 2019

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_