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Molien series for 3-dimensional group [2,7] = *227.
1

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

%S 1,0,2,0,3,0,4,1,5,2,6,3,7,4,9,5,11,6,13,7,15,9,17,11,19,13,21,15,24,

%T 17,27,19,30,21,33,24,36,27,39,30,42,33,46,36,50,39,54,42,58,46,62,50,

%U 66,54,70,58,75,62,80,66,85,70,90,75,95,80,100,85,105,90,111,95,117,100,123

%N Molien series for 3-dimensional group [2,7] = *227.

%H G. C. Greubel, <a href="/A008721/b008721.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=224">Encyclopedia of Combinatorial Structures 224</a>

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

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1,0,0,1,0,-2,0,1).

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

%p 1/((1-x^2)^2*(1-x^7));seq(coeff(series(%, x, n+1), x, n), n = 0..80); # modified by _G. C. Greubel_, Sep 09 2019

%t LinearRecurrence[{0,2,0,-1,0,0,1,0,-2,0,1}, {1,0,2,0,3,0,4,1,5,2,6}, 80] (* _G. C. Greubel_, Sep 09 2019 *)

%o (PARI) my(x='x+O('x^80)); Vec(1/((1-x^2)^2*(1-x^7))) \\ _G. C. Greubel_, Sep 09 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)^2*(1-x^7)) )); // _G. C. Greubel_, Sep 09 2019

%o (Sage)

%o def A008721_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 1/((1-x^2)^2*(1-x^7)) ).list()

%o A008721_list(80) # _G. C. Greubel_, Sep 09 2019

%o (GAP) a:=[1,0,2,0,3,0,4,1,5,2,6];; for n in [12..80] do a[n]:=2*a[n-2]-a[n-4]+a[n-7] -2*a[n-9]+a[n-11]; od; a; # _G. C. Greubel_, Sep 09 2019

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Terms a(60) onward added by _G. C. Greubel_, Sep 09 2019