A045513 Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)*(1-x^6)).

1, 1, 3, 5, 10, 15, 27, 39, 63, 90, 135, 187, 270, 364, 505, 670, 902, 1173, 1545, 1976, 2550, 3218, 4081, 5083, 6357, 7825, 9659, 11772, 14366, 17342, 20956, 25080, 30031, 35667, 42357, 49945, 58881

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

3

|

2--4--6--5--4--3--2--1

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Arjeh M. Cohen and Robert L. Griess Jr., On finite simple subgroups of the complex Lie group of type E_8, 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.
• Kaiwen Sun and Haowu Wang, Weyl invariant E8 Jacobi forms and E-strings, arXiv:2109.10578 [math.NT], 2021. See Table 1 p. 9.
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

For G2, the corresponding sequence is A001399.

For D4, the corresponding sequence is A001752.

For F4, the corresponding sequence is A115264.

For E6, the corresponding sequence is A164680.

For E7, the corresponding sequence is A210068.

Programs

Magma

R:=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

Maple

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

Mathematica

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 *)

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

Formula

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