A005868 - OEIS

%I #40 Sep 08 2022 08:44:34

%S 1,0,1,0,2,0,2,0,3,0,4,0,5,0,6,1,7,1,8,2,10,2,11,3,13,4,14,5,16,6,18,

%T 7,20,8,22,10,24,11,26,13,29,14,31,16,34,18,36,20,39,22,42,24,45,26,

%U 48,29,51,31,54,34,58,36,61,39,65,42,68,45,72,48,76,51

%N Molien series for 3-dimensional representation of Z2 X (double cover of A6), u.g.g.r. # 27 of Shephard and Todd.

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

%H Vincenzo Librandi, <a href="/A005868/b005868.txt">Table of n, a(n) for n = 0..1000</a>

%H G. C. Shephard and J. A. Todd, <a href="http://dx.doi.org/10.4153/CJM-1954-028-3">Finite unitary reflection groups</a>, Canadian J. Math. 6, (1954). 274--304. MR0059914 (15,600b).

%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 (0,1,0,1,1,-1,-1,0,-1,0,1).

%F G.f.: (1-x+x^2)*(1+x-x^3-x^4-x^5+x^7+x^8)/((1-x)^3*(1+x)^2*(1+x^2)*(1+x+x^2+x^3+x^4)). - _Colin Barker_, Jan 08 2014

%F a(n) ~ 1/80*n^2. - _Ralf Stephan_, Apr 29 2014

%F a(n) = a(n-2)+a(n-4)+a(n-5)-a(n-6)-a(n-7)-a(n-9)+a(n-11). - _Wesley Ivan Hurt_, May 24 2021

%p (1+x^45)/(1-x^6)/(1-x^12)/(1-x^30):

%p seq(coeff(series(expand(%), x, 3*n+1), x, 3*n), n=0..100);

%t CoefficientList[Series[(1-x+x^2)(1+x-x^3-x^4-x^5+x^7+x^8)/((1-x)^3 (1+x)^2 (1+x^2)(1+x+x^2+x^3+x^4), {x, 0, 70}], x]] (* _Vincenzo Librandi_, Apr 29 2014 *)

%t LinearRecurrence[{0,1,0,1,1,-1,-1,0,-1,0,1},{1,0,1,0,2,0,2,0,3,0,4},100] (* _Harvey P. Dale_, Aug 29 2016 *)

%o (PARI) Vec((x^10-x^5+1)/(-x^11+x^9+x^7+x^6-x^5-x^4-x^2+1) + O(x^100)) \\ _Colin Barker_, Jan 08 2014

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1+x^15)/((1 - x^2)*(1-x^4)*(1-x^10)) )); // _G. C. Greubel_, Feb 06 2020

%o (Sage)

%o def A005868_list(prec):

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

%o     return P( (1+x^15)/((1-x^2)*(1-x^4)*(1-x^10)) ).list()

%o A005868_list(65) # _G. C. Greubel_, Feb 06 2020

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_