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Molien series for 6-dimensional complex representation of double cover of J2.
1

%I #21 Sep 08 2022 08:44:34

%S 1,0,0,0,0,0,1,0,0,0,1,0,2,0,1,2,3,1,4,2,5,5,7,4,10,8,12,12,16,13,24,

%T 21,27,27,35,34,48,45,54,57,72,70,90,88,104,112,132,132,159,162,188,

%U 199,228,230,270,281,316,333,373,384,441,458,506,532,590,613

%N Molien series for 6-dimensional complex representation of double cover of J2.

%D J. H. Conway and N. J. A. Sloane, circa 1977.

%H Ray Chandler, <a href="/A005813/b005813.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_45">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 1, 1, 0, 1, 0, 0, -1, -1, -1, 0, -1, 0, 1, 0, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 1, 0, -1, 0, -1, -1, -1, 0, 0, 1, 0, 1, 1, 0, 0, -1).

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

%F a(n) ~ 1/2268000*n^5 + 1/151200*n^4 + 17/453600*n^3 (sequence without interleaved zeros). - _Ralf Stephan_, Apr 29 2014

%F G.f.: (1 -x^3 -x^4 +x^10 +x^11 +x^12 +x^16 -x^19 -x^23 +x^26 +x^30 +x^31 +x^32 -x^38 -x^39 +x^42)/((1-x^3)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^10)*(1-x^15)). - _G. C. Greubel_, Feb 06 2020

%p # p/q = 1 +x^12 +x^20 +2*x^24 +x^28 +..., where

%p p := x^140 +x^110 +x^108 +x^106 +2*x^104 +2*x^102 +3*x^100 +3*x^98 +3*x^96 +3*x^94 +4*x^92 +4*x^90 +4*x^88 +4*x^86 +4*x^84 +4*x^82 +4*x^80 +4*x^78 +3*x^76 +4*x^74 +3*x^72 +4*x^70 +3*x^68 +4*x^66 +3*x^64 +4*x^62 +4*x^60 +4*x^58 +4*x^56 +4*x^54 +4*x^52 +4*x^50 +4*x^48 +3*x^46 +3*x^44 +3*x^42 +3*x^40 +2*x^38 +2*x^36 +x^34 +x^32 +x^30 +1;

%p q := (1-x^12)*(1-x^20)*(1-x^24)*(1-x^28)*(1-x^30)*(1-x^32);

%p seq(coeff(series(p/q, x, 2*n+1),x,2*n), n=0..60);

%t CoefficientList[Series[(1-x^3-x^4+x^10+x^11+x^12+x^16-x^19-x^23+x^26+x^30+x^31 +x^32-x^38-x^39+x^42)/((1-x^3)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^10)*(1-x^15)), {x, 0, 60}], x] (* _G. C. Greubel_, Feb 06 2020 *)

%o (PARI) Vec( (1-x^3-x^4+x^10+x^11+x^12+x^16-x^19-x^23+x^26+x^30+x^31 +x^32-x^38-x^39+x^42)/((1-x^3)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^10)*(1-x^15)) +O('x^60) ) \\ _G. C. Greubel_, Feb 06 2020

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x^3-x^4+x^10 +x^11+x^12+x^16-x^19-x^23+x^26+x^30+x^31 +x^32-x^38-x^39+x^42)/((1-x^3)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^10)*(1-x^15)) )); // _G. C. Greubel_, Feb 06 2020

%o (Sage)

%o def A005813_list(prec):

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

%o return P( (1-x^3-x^4+x^10+x^11+x^12+x^16-x^19-x^23+x^26+x^30+x^31 +x^32-x^38-x^39+x^42)/((1-x^3)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^10)*(1-x^15)) ).list()

%o A005813_list(60) # _G. C. Greubel_, Feb 06 2020

%K nonn,easy,nice

%O 0,13

%A _N. J. A. Sloane_