This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A045513 #38 Dec 12 2023 08:39:16
%S A045513 1,1,3,5,10,15,27,39,63,90,135,187,270,364,505,670,902,1173,1545,1976,
%T A045513 2550,3218,4081,5083,6357,7825,9659,11772,14366,17342,20956,25080,
%U A045513 30031,35667,42357,49945,58881
%N A045513 Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).
%C A045513 This is associated with the root system E8, and can be described using the additive function on the affine E8 diagram:
%C A045513 3
%C A045513 |
%C A045513 2--4--6--5--4--3--2--1
%H A045513 G. C. Greubel, <a href="/A045513/b045513.txt">Table of n, a(n) for n = 0..1000</a>
%H A045513 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 A045513 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 A045513 <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 A045513 G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).
%p A045513 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 A045513 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 A045513 (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 A045513 (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 A045513 For G2, the corresponding sequence is A001399.
%Y A045513 For D4, the corresponding sequence is A001752.
%Y A045513 For F4, the corresponding sequence is A115264.
%Y A045513 For E6, the corresponding sequence is A164680.
%Y A045513 For E7, the corresponding sequence is A210068.
%K A045513 nonn,easy
%O A045513 0,3
%A A045513 _N. J. A. Sloane_