login
Molien series for group [2,5]+ = 225.
1

%I #25 Sep 08 2022 08:44:36

%S 1,0,2,0,3,1,5,2,7,3,10,5,13,7,16,10,20,13,24,16,29,20,34,24,39,29,45,

%T 34,51,39,58,45,65,51,72,58,80,65,88,72,97,80,106,88,115,97,125,106,

%U 135,115,146,125,157,135,168,146,180,157,192,168,205,180,218,192,231,205,245,218,259

%N Molien series for group [2,5]+ = 225.

%H G. C. Greubel, <a href="/A008798/b008798.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_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1,1,0,-2,0,1).

%F G.f.: (1+x^6)/((1-x^2)^2*(1-x^5)).

%F a(n) = (17 + 6*n + 2*n^2 + 5*(-1)^n*(3 + 2*n) + 8*A080891(n+4))/40.

%p A080891 := proc(n) op((n mod 5)+1,[0,1,-1,-1,1]) ; end proc:

%p A008798 := proc(n) 17/40+3*n/20+n^2/20+(-1)^n*(3/8+n/4) +A080891(n+4)/5; end proc:

%t CoefficientList[Series[(1+x^6)/((1-x^2)^2*(1-x^5)), {x,0,70}], x] (* _G. C. Greubel_, Sep 12 2019 *)

%o (PARI) my(x='x+O('x^70)); Vec((1+x^6)/((1-x^2)^2*(1-x^5))) \\ _G. C. Greubel_, Sep 12 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^6)/((1-x^2)^2*(1-x^5)) )); // _G. C. Greubel_, Sep 12 2019

%o (Sage)

%o def A008798_list(prec):

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

%o return P((1+x^6)/((1-x^2)^2*(1-x^5))).list()

%o A008798_list(70) # _G. C. Greubel_, Sep 12 2019

%o (GAP) a:=[1,0,2,0,3,1,5,2,7];; for n in [10..70] do a[n]:=2*a[n-2]-a[n-4]+a[n-5]-2*a[n-7]+a[n-9]; od; a; # _G. C. Greubel_, Sep 12 2019

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Definition clarified by _N. J. A. Sloane_, Feb 02 2018

%E More terms added by _G. C. Greubel_, Sep 12 2019