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Coefficients in Molien series for 5-dimensional faithful representation of Horrocks-Mumford group G_{HM}.
1

%I #22 Oct 06 2015 04:01:48

%S 1,0,1,2,6,6,21,27,48,73,112,149,224,289,390,507,656,812,1037,1259,

%T 1545,1866,2245,2648,3158,3679,4300,4983,5757,6579,7554,8562,9713,

%U 10960,12339,13798,15463,17190,19111,21170,23409,25767,28392,31116,34092,37255,40654,44219,48119,52162,56523

%N Coefficients in Molien series for 5-dimensional faithful representation of Horrocks-Mumford group G_{HM}.

%H Colin Barker, <a href="/A258702/b258702.txt">Table of n, a(n) for n = 0..1000</a>

%H Yang-Hui He, John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1,-1,2,-3,3,-2,1,1,-1,1,-2,1).

%F The Molien series is (1 - 2*z^5 + 2*z^10 - z^15 + 4*z^20 - 4*z^25 + 12*z^30 - 9*z^35 + 12*z^40 - 4*z^45 + 4*z^50 - z^55 + 2*z^60 - 2*z^65 + z^70)/((1 - z^30)*(1 - z^20)*(1 - z^15)*(1 - z^5)^2); the terms for n != 0 mod 5 are omitted in this sequence.

%F G.f.: -(x^14 -2*x^13 +2*x^12 -x^11 +4*x^10 -4*x^9 +12*x^8 -9*x^7 +12*x^6 -4*x^5 +4*x^4 -x^3 +2*x^2 -2*x +1) / ((x -1)^5*(x +1)^2*(x^2 -x +1)*(x^2 +1)*(x^2 +x +1)^2). - _Colin Barker_, Oct 05 2015

%t CoefficientList[Series[(x^14 - 2 x^13 + 2 x^12 - x^11 + 4 x^10 - 4 x^9 + 12 x^8 - 9 x^7 + 12 x^6 - 4 x^5 + 4 x^4 - x^3 + 2 x^2 - 2 x + 1)/((1 - x)^5 (x + 1)^2 (x^2 - x + 1) (x^2 + 1) (x^2 + x + 1)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 06 2015 *)

%o (PARI) Vec(-(x^14 -2*x^13 +2*x^12 -x^11 +4*x^10 -4*x^9 +12*x^8 -9*x^7 +12*x^6 -4*x^5 +4*x^4 -x^3 +2*x^2 -2*x +1) / ((x -1)^5*(x +1)^2*(x^2 -x +1)*(x^2 +1)*(x^2 +x +1)^2) + O(x^100)) \\ _Colin Barker_, Oct 05 2015

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_, Jun 08 2015, based on an email from _Yang-Hui He_.