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Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).
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%I #38 Dec 12 2023 08:39:16

%S 1,1,3,5,10,15,27,39,63,90,135,187,270,364,505,670,902,1173,1545,1976,

%T 2550,3218,4081,5083,6357,7825,9659,11772,14366,17342,20956,25080,

%U 30031,35667,42357,49945,58881

%N Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).

%C This is associated with the root system E8, and can be described using the additive function on the affine E8 diagram:

%C 3

%C |

%C 2--4--6--5--4--3--2--1

%H G. C. Greubel, <a href="/A045513/b045513.txt">Table of n, a(n) for n = 0..1000</a>

%H Arjeh M. Cohen and Robert L. Griess Jr., <a href="https://research.tue.nl/en/publications/on-finite-simple-subgroups-of-the-complex-lie-group-of-type-e8">On finite simple subgroups of the complex Lie group of type E_8</a>, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 367-405, Proc. Sympos. Pure Math., 47, Part 2, Amer. Math. Soc., Providence, RI, 1987.

%H Kaiwen Sun and Haowu Wang, <a href="https://arxiv.org/abs/2109.10578">Weyl invariant E8 Jacobi forms and E-strings</a>, arXiv:2109.10578 [math.NT], 2021. See Table 1 p. 9.

%H <a href="/index/Rec#order_30">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,0,-1,-4,-1,0,3,6,1,0,-4,-5,-5,0,5,5,4,0,-1,-6,-3,0,1,4,1,0,-2,-1,1).

%F G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).

%p seq(coeff(series(1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Jan 13 2020

%t CoefficientList[Series[1/((1-x)(1-x^2)^2(1-x^3)^2(1-x^4)^2(1-x^5)(1-x^6)),{x,0,40}],x] (* _Harvey P. Dale_, Sep 16 2019 *)

%o (PARI) Vec(1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)) )); // _G. C. Greubel_, Jan 13 2020

%Y For G2, the corresponding sequence is A001399.

%Y For D4, the corresponding sequence is A001752.

%Y For F4, the corresponding sequence is A115264.

%Y For E6, the corresponding sequence is A164680.

%Y For E7, the corresponding sequence is A210068.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_