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

%I #18 Sep 08 2022 08:44:36

%S 1,0,2,0,3,0,4,0,5,1,6,2,7,3,8,4,10,5,12,6,14,7,16,8,18,10,20,12,22,

%T 14,24,16,27,18,30,20,33,22,36,24,39,27,42,30,45,33,48,36,52,39,56,42,

%U 60,45,64,48,68,52,72,56

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

%H G. C. Greubel, <a href="/A008736/b008736.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_19">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1).

%F G.f.: (1+x^9)/((1-x^2)^2*(1-x^16)).

%p seq(coeff(series((1+x^9)/((1-x^2)^2*(1-x^16)), x, n+1), x, n), n = 0 .. 70); # modified by _G. C. Greubel_, Jul 30 2019

%t CoefficientList[Series[(1+x^9)/((1-x^2)^2*(1-x^16)), {x,0,70}], x] (* _G. C. Greubel_, Jul 30 2019 *)

%t LinearRecurrence[{1,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1},{1,0,2,0,3,0,4,0,5,1,6,2,7,3,8,4,10,5,12},70] (* _Harvey P. Dale_, Oct 03 2020 *)

%o (PARI) my(x='x+O('x^70)); Vec((1+x^9)/((1-x^2)^2*(1-x^16))) \\ _G. C. Greubel_, Jul 30 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^9)/((1-x^2)^2*(1-x^16)) )); // _G. C. Greubel_, Jul 30 2019

%o (Sage) ((1+x^9)/((1-x^2)^2*(1-x^16))).series(x, 70).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 30 2019

%o (GAP) a:=[1,0,2,0,3,0,4,0,5,1,6,2,7,3,8,4,10,5,12];; for n in [20..70] do a[n]:=a[n-1]+a[n-2]-a[n-3]+a[n-16]-a[n-17]-a[n-18]+a[n-19]; od; a; # _G. C. Greubel_, Jul 30 2019

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_